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Transmission Management of Delay-SensitiveMedical Packets in Beyond Wireless Body Area

Networks: A Queueing Game ApproachChangyan Yi, Student Member, IEEE , and Jun Cai, Senior Member, IEEE

Abstract—In this paper, the management of delay-sensitive medical packet transmissions in beyond wireless body area networks(beyond-WBANs) is studied. The considered system addresses the random arrival of sensed medical packets at each WBAN-gateway,which are categorized into different classes (one class of emergent alarms and multiple classes of non-emergent routines). Uponreceiving a medical packet, the associated gateway immediately declares a beyond-WBAN transmission request to the base station(BS). With the consideration of medical-grade quality of service (mQoS) requirements, the beyond-WBAN transmissions ofheterogeneous packets are scheduled by following the constructed queueing models with specifically designed priority disciplines. Byfurther considering the potential strategic behaviors of smart gateways, a non-cooperative delay-dependent prioritized queueing gamefor the beyond-WBAN transmission management is formulated. After that, we propose a novel analytical framework to jointlycharacterize the queueing performance and the properties of the game equilibrium. Theoretical and simulation results justify thefeasibility and applicability of our designed transmission management system in beyond-WBANs.

Index Terms—Beyond-WBANs, mQoS, queueing scheduling, delay-dependent priority, non-cooperative game.

F

1 INTRODUCTION

NOWDAYS, our ways of living are under gradual butcontinuous change with the ever-advancing science

and technology. These include the rapid development ofclinical medicine, immunology, nutrition, etc., which largelyprolong human life span and increase demands for high-quality healthcare services. However, the traditional med-ical facilities can hardly meet future needs because of thelimitations on hospital capacity and medical workforce [1].This dilemma necessitates the implementation of pervasivehealthcare monitoring systems, which integrate signal pro-cessing, computer engineering and wireless technologies, toachieve remote information collection, provide early detec-tion of diseases, and improve independence of patients.

Wireless body area networks (WBANs) are fundamentalcomponents for the realization of pervasive healthcare mon-itoring. A WBAN typically consists of several low-powerbiosensors for monitoring physiological data that profilehealth status of human body, and a gateway that aggregatesthe sensing data from biosensors and forwards them toremote medical centers via base stations (BSs) for interpre-tation. As a promising paradigm, designing WBAN-basedhealthcare monitoring systems has become increasinglypopular in both academia and industry [2], [3]. However,compared to the widely studied intra-WBAN communica-tions (the information exchange between biosensors and thegateway), the management of beyond-WBAN transmissions(the medical packet deliveries from the gateway to remotemedical centers), though of equal importance and necessity,

• C. Yi and J. Cai are with the Department of Electrical and ComputerEngineering, University of Manitoba, Winnipeg, MB, Canada, R3T 5V6.(E-mail: changyan.yi@umanitoba.ca, jun.cai@umanitoba.ca).

is only recently attracting research attentions [4], [5]. In fact,because of the following reasons, it is expected that the radiobandwidth available for beyond-WBANs will be saturated.

• Wi-Fi (or any other technologies working on low-frequency bandwidths) is not suitable for beyond-WBAN communications due to its considerably re-stricted radio coverage [6]. Commonly, electronichealth (e-health) systems require pervasive and ubiq-uitous monitoring so that beyond-WBAN access at“anywhere” and “anytime” should be guaranteed.Any service interruption or packet loss may lead toserious consequences in healthcare.

• Cellular-like networks is more preferable for beyond-WBAN communications. However, traditional cellu-lar systems have already been crowded with a largenumber of subscribed mobile users, and the ever-increasing demand in cellular access is rapidly strain-ing the capacity of existing cellular networks. Thus,only very limited radio bandwidth can be allocatedfor beyond-WBAN communications.

• In addition, with the development of e-health tech-nology, it has been envisioned that there will be alarge deployment of WBANs in a short future [7].This may further lead to a heavy burden on radioresource management for beyond-WBANs.

Moreover, unlike conventional wireless networks, oneunique feature of beyond-WBAN transmissions is the re-quirement of guaranteeing medical-grade quality of ser-vice (mQoS) provisioning. Specifically, mQoS requires thati) emergent medical situations have to be reported in ahigher priority over those with regular importance [8], andii) transmission services for non-emergent medical signalsshould also be differentiated by their heterogeneous delay

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sensitivities with respect to different application purposes.For instance, it is intuitive that the data of electrocardiogram(ECG) is more critical than those of temperature, eventhough both of them are under non-emergency. Applyingan absolute priority rule as discussed in our previous works[4], [9], [10] can maintain the transmission priorities amongdifferent medical criticality levels, but such rule may leadto tremendously large waiting delays for less importantpackets. However, in practice, these “less important” medi-cal packets are also critical components of patients’ healthprofiles and are important in improving the accuracy ofdiagnosing [11]. Thus, excessive delays in their transmis-sions may at the end deteriorate health services to patients.Therefore, when designing protocols for managing beyond-WBAN transmissions, the original packet criticality and theexperienced waiting delays have to be considered jointly.

Furthermore, as widely discussed in the WBAN-relatedliterature [7], [12], gateways in beyond-WBANs are com-monly smart phones or any other smart devices carried bypatients for medical data aggregation, processing and trans-mission. Therefore, WBAN-gateways are intelligent enough(due to the rapid development of device intelligence) to po-tentially behave selfishly and strategically in order to benefitthemselves. Such game-theoretic feature in medical packettransmission scheduling has already been discussed in somerecent works [5], [13], [14]. Hence, it is intuitive that multiplegateways are going to compete with each other for thelimited beyond-WBAN resources to transmit different med-ical packets collected from their own associated biosensors.Meanwhile, the BS is responsible to determine appropriatescheduling schemes for packet transmissions from differentgateways with the guarantee of mQoS provisioning.

Obviously, designing such beyond-WBAN transmissionmanagement systems by considering all aforementionedconcerns is quite challenging due to the following aspects:

a. The beyond-WBAN transmission scheduling should beable to i) protect emergent information delivery, andii) adopt a dynamic priority rule for transmitting non-emergent medical packets, which depends on the originalpacket criticality and at the same time varies with theirexperienced waiting delays. This requires the applicationof system dynamics at packet level.

b. Each gateway is strategic about the mQoS for its packettransmissions, while such mQoS is in turn determinedby the transmission scheduling mechanism and all othergateways’ strategies. Due to the implicit relationship be-tween the transmission scheduling of the beyond-WBANand the strategies of gateways, the packets’ delay andequilibrium performance have to be jointly analyzed.

To tackle these difficulties, in this paper, we propose anovel queueing game based transmission management sys-tem for delay-sensitive medical packets in beyond-WBANs.In the considered model, sensed medical packets arrive ran-domly at each gateway via intra-WBAN communications,and all packets are categorized into different classes (whichconsists of one class of emergent alarms and multiple classesof non-emergent routines). Upon receiving a medical packet,each gateway immediately declares a beyond-WBAN trans-mission request to the BS and temporarily stores the packetin its own buffer before it is scheduled to be transmitted.

As the central controller, the BS determines a transmissionscheduling scheme based on the mQoS requirements. Thepacket-level operations of the beyond-WBAN managementare modeled as queueing systems with first-come first-serve(FCFS) and delay-dependent priority disciplines for emer-gent and non-emergent medical packet transmissions, re-spectively. By guaranteeing a satisfied QoS for all emergentinformation delivery and further considering the strategicbehaviors of gateways in reporting the specific medicalcriticality of their non-emergent packets, a non-cooperativedelay-dependent prioritized queueing game is formulated.After that, an analytical framework is proposed whichjointly characterizes the properties of the queueing systemand the game equilibrium. It is worth noting that the pro-posed beyond-WBAN transmission scheduling frameworkcan be potentially implemented with the help of upcomingnetwork slicing techniques [15], by which various wirelessresources required for beyond-WBAN transmissions canbe virtualized as servers in queueing systems so that thephysical-layer resource allocations can be performed by thenetwork-slicing based scheduling management.

The main contributions of this paper are as follows:

• The dynamic scheduling of beyond-WBAN medicalpacket transmissions is modeled by M/G/K queueswith a generally distributed service (transmission)time and specifically designed priority disciplines.

• A non-cooperative queueing game is formulated todeal with the impact of gateways’ strategic behaviorsin the beyond-WBAN transmission management.

• The waiting delays of medical packet transmissionsin the delay-dependent prioritized M/G/K queueis expressed by a recursive function in terms ofgateways’ strategies.

• The equilibrium of the formulated game is studiedwith a consideration of the delay-dependent priori-tized queueing performance.

• Both theoretical and simulation results are demon-strated to examine the feasibility and applicabilityof our proposed management framework in beyond-WBAN transmissions.

The rest of this paper is organized as follows: Section2 gives a review of related works. Section 3 introduces theconsidered model and shows the problem description. Sec-tion 4 presents the formulated delay-dependent prioritizedqueueing game. Simulation results are illustrated in Section5. Finally, a brief conclusion is summarized in Section 6.

2 RELATED WORKS

As one of the most important features of healthcare monitor-ing services, mQoS have been widely explored in WBAN-related designs and managements. Moulik et al. in [16]presented an adaptive access control algorithm for enhanc-ing the transmission reliability of critical sensor nodes inWBANs. Manirabona et al. in [17] proposed a priority-weighted round robin transmission scheduling which aimedto minimize the delay of emergent traffic flows. A coopera-tive game-theoretic bargaining approach for priority-baseddata-rate tuning among biosensors in WBANs was dis-cussed in [18], where the QoS of critical information delivery

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was improved. However, these works limited their focuseson intra-WBAN communications only. The authors in [10]investigated a resource allocation problem for transmittingprioritized medical packets in beyond-WBANs, where anon-atomic game was formulated. In [4], Yi et al. introducedan incentive-compatible mechanism for managing medicalpackets’ transmissions according to an absolute priorityrule. However, none of these works addressed the issue thathealth conditions may deteriorate due to excessive delays ofnon-emergent medical packets, which requires transmissionpriorities of different packets to be changed dynamically.

Game-theoretic techniques [19] have been employed invarious wireless networks and applications [20]–[23] for an-alyzing the equilibrium among multiple strategic and selfishusers, and queueing modeling [24] is a natural and effectiveway to describe the transmission scheduling, and queueinggame has also been studied recently. In [25], Sarikaya etal. built up a dynamic pricing game for uplink randomaccess in contention-based wireless networks, where users’queueing stabilities was taken into account. In [26], a dis-tributed queueing game in an interference-limited wirelessnetwork was designed, where each user could individuallydetermine whether to enqueue a packet or transmit it to thedestination. However, the models described in these worksdo not fit the practical healthcare management due to thelack of considerations on mQoS for heterogeneous medicalpacket transmissions.

In summary, this paper differs from all existing works inthe literature by considering the following issues specifiedto practical pervasive healthcare monitoring systems.

• Instead of intra-WBANs, the management of medicalpacket transmissions in beyond-WBANs is studied.

• Besides guaranteeing QoS provisioning for emer-gent medical packets, a dynamic delay-dependentpriority discipline is adopted for scheduling non-emergent packet transmissions.

• Medical packets are characterized into differentclasses with heterogeneous arrival rates, delay sen-sitivities and transmission time.

• In order to model the strategic behaviors of gatewaysin beyond-WBANs, a delay-dependent prioritizedqueueing game is proposed and analyzed.

3 SYSTEM MODEL AND PROBLEM DESCRIPTION

In this section, we first characterize the network model ofbeyond-WBANs under consideration. Then, the schedulingmanagement for delay-sensitive beyond-WBAN medicalpacket transmissions is described in detail with the con-struction of a queueing game.

3.1 Network ModelAs illustrated in Fig. 1, a pervasive healthcare monitoringsystem is considered, which consists of the infrastructuresof intra-WBANs and beyond-WBANs. Each intra-WBAN isdeployed on a patient, and ordinarily includes a gateway(e.g., a smart phone) and a number of heterogeneous biosen-sors worn on different parts of the human body. Biosensorsare used to continuously monitor physiological signals andtransmit them to the gateway. This information exchange

Fig. 1. An illustration of pervasive healthcare monitoring networks.

is realized with the help of intra-WBAN communicationswhich have been defined in some existing standards, suchas IEEE 802.15.6 [27]. Each gateway, as a local processor,collects all medical signals from its associated biosensors,temporarily stores these data in its buffer (i.e., data stor-age), and then forwards them to the BS (which is furtherconnected to medical centers through Internet) via thebeyond-WBAN. Since our emphasis of this paper is on thescheduling management of medical packet transmissions inthe beyond-WBAN, the intra-WBAN communications areomitted, and we refer the interested readers to [7] for detailson intra-WBAN design. Note that pervasive healthcare sys-tems may also raise various security and privacy issues [28],[29]. Since healthcare information is relatively sensitive forpatients, any unintended disclosure may violate patients’privacy. Moreover, without appropriate security protections,malicious gateways or attackers may intentionally manipu-late the medical packet transmission scheduling, leading toserious consequences in healthcare. The design of securityand privacy-preserving mechanisms is out of the scope ofthis paper, and we refer interested readers to [30], [31] forrecent advances in this area.

For explanation purpose, we consider a cellular-likebeyond-WBAN architecture [4] with a single BS1 and Ngateways (each of which stands for one patient). The BSowns K homogeneous orthogonal radio channels that arefixed and dedicated for remote healthcare services, and isresponsible to the management of beyond-WBAN medicalpacket transmissions. Each gateway may receive a varietyof medical packets that are either emergent alarms or non-emergent routines (which may belong to different medicalapplications, e.g., EEG, ECG and EMG) from its associatedbiosensors. To be consistent with the IEEE standard [27]for intra-WBAN communications, in this paper, all medicalpackets are categorized into a discrete set of classes, i.e.,L = {0, 1, 2, . . . , L}, where {0} and {1, 2, . . . , L} representthe sets of classes for emergent alarms and all non-emergentroutines, respectively. In practice, some existing methods,such as the generalized fuzzy inference-based technique[33], can be applied to classify medical packets based ontheir severity. Since medical packets/signals within thesame class are similar in terms of application purpose,medical criticality, and sensing frequency, without loss of

1. For any network with multiple BSs, an additional BS associationalgorithm [32] can be adopted to decouple the network into multiplesubnetworks, where each of them has a single BS.

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generality, the size (in bits) of medical packets in each class`,∀` ∈ L, is modeled as a random variable S` followinga general distribution with a probability density function(PDF) fS`

(·) and a finite mean E[S`].Each gateway is allowed to transmit its medical packets

to the BS on different and possibly multiple channels. Thus,the beyond-WBAN transmission rate of each gateway n onany one channel can be written as [34]

rn = B log

(1 +|hn|2Pd−αn

σ2

), (1)

where B, P and σ2 stand for the channel bandwidth, thetransmission power and the additive white Gaussian noisevariance, respectively; |hn|2 captures the Rayleigh fadingeffect and is commonly modeled by an exponential randomvariable with a unit mean; dn is the distance betweengateway n and the BS, so that d−αn signifies the path losseffect, where the path loss exponent α ≥ 2 [34].

3.2 Queueing SchedulingIn pervasive healthcare monitoring networks, each WBANconsists of up to 256 biosensors deployed on a patient[27], and it is expected that with the future advancementsin lightweight sensors and the low-power transmissiontechnology, the number of biosensors associated with aWBAN may even increase for fulfilling more comprehensiveand accurate healthcare monitoring [7]. Thus, the aggregatearrival of medical packets from a relatively large num-ber of biosensors at each gateway n (i.e., receiving fromintra-WBAN communications) can be well approximatedby a Poisson process with an average rate λn. Similarassumption on Poisson/Markovian medical packet arrivalprocess has also been employed in [6], [35]. However,the proposed analytical framework is applicable to moregeneral arrival processes. Besides, with long-term healthcondition tracking, there is a known probability distributionPn = (Pn,0, Pn,1, . . . , Pn,L) at each gateway n, where Pn,`represents the probability that an arrived packet at gatewayn is in class `,∀` ∈ L, and obviously

∑L`=0 Pn,` = 1.

Notice that in a practical healthcare system, the probabilitiesof having emergent medical signals are much smaller thanthose of normal ones, i.e., Pn,0 � Pn,`,∀` ∈ L\{0}. GivenPn,∀n ∈ {1, . . . , N}, the average arrival rate of the `thclass medical packets at gateway n can be calculated asλnPn,`,∀` ∈ L. Whenever a gateway receives a medicalpacket from any of its biosensors, it will immediately declarea beyond-WBAN transmission request to the BS. All medicalpackets are temporarily stored in gateways’ buffers beforethey are completely transmitted. In this paper, we do notconsider buffer overflow2. Fig. 2 shows the queueing modelof the beyond-WBAN transmission management.

As the central controller in the beyond-WBAN, the BS isresponsible for scheduling the transmissions of all medicalpackets from all gateways within its cell. Since the beyond-WBAN transmission services should only be differentiatedby packets’ medical criticality, rather than the associatedgateways where these packets are originated, the BS can

2. Since current smart devices commonly come with multi-gigabytestorages while medical packets are usually in hundreds kilobits, theeffect of buffer overflow is negligible.

λ1P1,0

λ1P1,L

λ2P2,0

λ2P2,L

λNPN,0

λNPN,L

Λ0 =

N∑

n=1

λnPn,0

Λ1 =

N∑

n=1

λnPn,1

ΛL =

N∑

n=1

λnPn,L

...

...

...

...

...Base station

Channel 1

Channel 2

Channel K

Beyond-WBAN

transmissions

Arrival of

medical packets

Virtual gateway

Gateway 1

Gateway 2

Gateway N

...

Emergent alarms

Non-emergent routines

Fig. 2. Queueing model of beyond-WBAN packet transmissions.

treat all packet transmission requests from a single virtualgateway, as depicted in Fig. 2, with two specific virtualbuffers, i.e., one for emergent alarms and the other for allnon-emergent routines. The arrival of medical packets at thevirtual gateway consists of L+ 1 different arrival processeswith regard to L+1 packet classes. Since gateways are inde-pendent with each other, the arrivals of packet transmissionrequests at the virtual gateway are still Poisson processes,and the average packet arrival rates can be calculated as

Λ` =N∑

n=1

λnPn,`, ∀` ∈ L. (2)

Furthermore, consider that all gateways are randomlydistributed in the cell of the BS, e.g., Poisson point distri-bution as in [36], [37]. Then, from the view of the centralcontroller (i.e., BS), the beyond-WBAN transmission rate ofall medical packets (regardless of particular gateways) canbe derived and generalized as a random variable R with aPDF fR(·) and a finite mean E[R] based on the expressionof individual transmission rate shown in (1).

Unlike most of the conventional wireless applicationsthat aim to maximize network throughputs, the objectiveof beyond-WBAN is to deliver heterogeneous medical datato the BS in a timely manner. This is because all medicalpackets are sensitive to the potential waiting delays intransmissions [38] (such delay-sensitivity will be furthercharacterized in Section 3.3). Besides, medical applicationsalways require that the service for emergent illness has tobe protected in a higher priority over those with regu-lar importance [8]. In other words, non-emergent medicalpackets (that are in the queue of non-emergent routines)can be served only if the QoS requirement of transmittingemergent packets (that are in the queue of emergent alarms)has already been met, i.e.,

E[W0] ≤ γth, (3)

where E[W0] is the expected waiting delay for emergentpacket transmissions in the beyond-WBAN, and γth rep-resents a pre-determined QoS threshold. In order to guar-antee the satisfaction of condition (3), the BS can grant K0

channels for exclusively serving emergent packet transmis-sions. Note that K0 can be obtained according to differentQoS requirements. For explicit analysis, in this paper, the

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derivation of K0 is based on the mean delay requirement(i.e., the satisfaction of inequality (3)). In practice, K0 canalso be numerically calculated by imposing a requirementon higher-order statistics of the waiting delay for emergentpacket transmissions.

Then, the management of beyond-WBAN transmissionscheduling for emergent alarms can be modeled by a FCFSM/G/K0 queue with a Poisson arrival process at an averagerate Λ0 and a General service process in an average rateK0µ0, where

1/µ0 = E[S0/R] = E[S0]

∫ ∞

−∞

1

rfR(r)dr, (4)

is the average transmission time for one emergent medicalpacket on one channel (since S0 and R denote the packetsize and transmission rate, respectively). According to [39],the delay performance of M/G/K0 queue can be wellapproximated by

E[WM/G/K0 ] ≈(C2 + 1

2

)E[WM/M/K0 ], (5)

where C2 is the coefficient-of-variation of the service time,which can be calculated by

C2 =Var[S0/R]

(E[S0/R])2= µ2

0E[S20 ]

∫ ∞

−∞

1

r2fR(r)dr − 1, (6)

and E[WM/M/K0 ] denotes the mean waiting delay of acorresponding M/M/K0 queue with an exponential servicetime distribution. Thus, according to the queueing theory[40], we have

E[WM/M/K0 ] =Q(K0,ρ0)

µ0(K0 − ρ0), (7)

where ρ0 = Λ0/µ0 and

Q(K0,ρ0) =ρK0

0 /K0!

[(K0−ρ0)/K0]∑K0−1k=0 (ρk0/k!) + ρK0

0 /K0!. (8)

Substituting (4)-(8) to inequality (3), the minimum num-ber of channels, i.e., Kmin

0 , that has to be granted for pro-tecting emergency can be obtained as

Kmin0 = arg min

K0

E[WM/G/K0 ] ≤ γth. (9)

The numerical calculation and analysis of Kmin0 will be

presented in Section 5.Since the traffic of emergent alarms is relatively low

(because Pn,0 � Pn,`,∀` ∈ L\{0}), we can assume thatKmin

0 < K always holds. Then, the beyond-WBAN trans-mission management for non-emergent medical packets canbe modeled by another queueing system, in which theservice time for each non-emergent medical packet in class`,∀` ∈ L\{0}, can be expressed as

T` =S`

R(K −Kmin0 )

, ∀` ∈ L\{0}, (10)

where K − Kmin0 is the remaining number of channels

available for non-emergent packet transmissions, and thuswe have

E[T`] =E[S`]

∫∞−∞

1rfR(r)dr

(K −Kmin0 )

, ∀` ∈ L\{0}; (11)

E[T 2` ] =

E[S2` ]∫∞−∞

1r2 fR(r)dr

(K −Kmin0 )2

, ∀` ∈ L\{0}. (12)

Even though transmission requests of all non-emergentmedical packets are considered to be stored and served ina same virtual buffer with unprotected QoS, the services(i.e., transmission orders) of these packets in the beyond-WBAN are further differentiated depending on factors, suchas the medical application-specific packet criticality andthe experienced waiting delay in the buffer, which will beexplained in detail in the next subsection.

3.3 Game FormulationNon-emergent medical packets are also delay-sensitive ow-ing to the common feature of medical applications, i.e., themedical value of one packet will decrease with the increaseof delay in transmissions. In other words, the potentialwaiting delays in beyond-WBAN transmissions will incurwaiting costs on the values of all non-emergent medicalpackets. By taking into account the heterogeneities amongdifferent class of medical packets, we introduce θ` as the costper unit of waiting delay for each non-emergent medicalpacket in class `,∀` ∈ L\{0}, which also represents the `thclass packets’ delay sensitivity. Without loss of generality,we assume that

θ1 ≥ θ2 ≥ . . . ≥ θL. (13)

In practice, such packet delay sensitivities are related todifferent medical applications and can be pre-estimated bymedical specialists from empirical measurements.

Since the BS is unaware of the specific medical sta-tus/information of each individual gateway, when report-ing a beyond-WBAN transmission request for any non-emergent medical packet to the BS, each gateway is requiredto declare an additional packet criticality coefficient, which isdefined as the importance of this packet to the gateway.Then, BS can schedule the beyond-WBAN transmissionsbased on a decreasing order of its calculated delay-dependentpriorities (DPs) of all buffered non-emergent packets, i.e.,medical packets with higher DPs will be scheduled to trans-mit prior to those with lower DPs.Definition 1 (DP). For any non-emergent medical packet in

class `,∀` ∈ L\{0}, which arrived at the gateway n attime τ , its delay-dependent priority at time t,∀t ≥ τ ,is calculated as βn,`(t − τ), where βn,` ∈ [0,∞) is thecriticality coefficient declared by gateway n.

According to this definition, DP is as a linear function ofthe criticality coefficient and the experienced waiting delay.However, the formulation of DP is actually not restrictedto linear functions, and the only necessary requirementis the guarantee of a “class-dependent priority”, i.e., i.e.,medical packets in a class with higher emergency shouldalways have higher priorities than the ones in other classeswith less importance, if they have experienced the samewaiting delays. In fact, this is a fundamental feature forhealthcare monitoring because medical packets are origi-nally categorized into different classes according to existingIEEE standard [27] based on their medical criticality. In thispaper, we take the linear form of DP as an example toshow the analysis procedure, while any other forms, which

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meet “class-dependent priority”, can also be employed withslight mathematical modifications.

Considering that all gateways are intelligent and ra-tional, each gateway n is able to strategically determineβn,` for each medical packet in class `,∀` ∈ L\{0}, withthe objective of minimizing its expected cost in beyond-WBAN transmissions. Here, the total cost for transmittingone non-emergent medical packet with βn,` by gateway n isformulated as

Cn,` = θ`E[W (βn,`)] + π(βn,`), ∀` ∈ L\{0}, (14)

where E[W (βn,`)] is the mean waiting delay, and thusθ`E[W (βn,`)] represents the mean waiting cost3; π(βn,`)denotes the service charge by the BS for its beyond-WBANtransmission. Note that both E[W (βn,`)] and π(βn,`) arefunctions of βn,`.

In order to minimize Cn,`,∀` ∈ L\{0},∀n ∈ {1, . . . , N},all gateways will compete with each other by adjustingβn,`,∀` ∈ L\{0},∀n ∈ {1, . . . , N}, so as to possibly de-crease their packets’ expected waiting delays in beyond-WBAN transmission scheduling while at the same time low-ering the service charges. This obviously results in a non-cooperative game on a delay-dependent prioritized queue-ing system. Unlike the traditional non-cooperative gamemodels [19], analyzing such delay-dependent prioritized queue-ing game is much more challenging because it is difficult,if not impossible, to explicitly characterize the relationshipbetween the strategy (i.e., βn,`) and the corresponding delayperformance (i.e., E[W (βn,`)]). In the following sections, wewill propose a new analytical framework to jointly study thequeueing and equilibrium performances of this formulateddelay-dependent prioritized queueing game.

For convenience, the timeline of the beyond-WBANtransmission management is summarized as follows, andTable 1 lists some important notations used in this paper.

• Based on mQoS requirements and health informationstatistics, the BS calculates and assigns Kmin

0 andK − Kmin

0 channels for supporting emergent andnon-emergent beyond-WBAN medical packet trans-missions, respectively.

• The BS broadcasts the scheduling scheme, i.e., FCFSfor emergent alarms and delay-dependent priori-tized transmission order for non-emergent routines.

• Whenever a gateway receives a medical packetfrom the intra-WBAN, it will immediately declare abeyond-WBAN transmission request to the BS alongwith a strategically determined criticality coefficientfor each non-emergent medical packet.

• The BS manages the beyond-WBAN packet trans-missions online according to the pre-determinedscheduling scheme.

Notice that, following the similar discussions as in [6],[9], control signalling (including the report of transmissionrequests and the manage of transmission scheduling) istransmitted via a dedicated common control channel, and itspotential overhead is ignored since it is negligible comparedto regular medical packet transmissions.

3. To match the structure of the delay-dependent priority defined inDefinition 1, the mean waiting cost is also defined as a linear function.

TABLE 1IMPORTANT NOTATIONS IN THIS PAPER

Symbol MeaningN number of WBAN gatewaysK number of channelsKmin

0 minimum number of channels for emergencyL set of all medical packet classes{0} set of class for emergent alarmsL\{0} set of classes for non-emergent routinesS`, fS`

(·) size of medical packets in class ` and its PDFR, fR(·) beyond-WBAN transmission rate and its PDFΛ` aggregate arrival rate of packets in class `T` service time of non-emergent packets in class `E[W (·)] mean delay in beyond-WBAN transmissionsθ` cost per unit of delay for packets in class `βn,` criticality coefficient stamped by gateway nβ` symmetric strategy for packets in class `β, β̂ pure (mixed) symmetric strategy profileβe, β̂e pure (mixed) symmetric Nash equilibriumCn,` cost for transmitting each packet by gateway nC` symmetric cost for transmitting each packet

4 ANALYSIS OF THE DELAY-DEPENDENT PRIORI-TIZED QUEUEING GAME

In this section, the structure of the delay-dependent prior-itized queueing game for beyond-WBAN medical packettransmissions is discussed. After that, the delay perfor-mance of the queueing system and the properties of thegame equilibrium are analyzed in detail.

4.1 Structure of the GameIn practical pervasive healthcare monitoring systems, it isexpected that the number of patients equipped with WBAN-gateways will become tremendously large [10]. As a result,the constructed delay-dependent prioritized queueing gamefor managing the beyond-WBAN transmissions of non-emergent medical packets can be considered to be non-atomic [41], [42] in the sense that the action of a singlesmart gateway has no impact on the steady-state proper-ties of the overall queueing system. This implies that allgateways face the same expected waiting delay and thecost function for transmitting medical packets in the sameclass `,∀` ∈ L\{0}, and thus have the same best responsesin determining the packet criticality coefficients. In otherwords, any equilibrium of this queueing game is symmetricwithin each class of medical packets among all gateways.

Hence, we can consider a symmetric mixed strategyprofile β̂ = {β̂`,∀` ∈ L\{0}}, where β̂` denotes the mixedstrategy on criticality coefficients employed by all gatewaysfor medical packets in class `,∀` ∈ L\{0}. Note that β̂`specifies a probability distribution on different values of β`.Then, the cost function for transmitting a medical packet inclass `,∀` ∈ L\{0}, with a criticality coefficient β` under β̂can be rewritten as

C`(β`, β̂) = θ`E[W (β`, β̂)] + π(β`). (15)

Intuitively, given a fixed β̂, medical packets with a largerβ` can obtain a better beyond-WBAN transmission service(i.e., a smaller E[W (·)]). However, a larger β` should alsolead to a higher service charge. To meet this intuition, wedefine π(β`) = ξβ`, where ξ is a system-determined factor(which without loss of generality, is normalized to 1 for

7

simplicity in the following analysis). Therefore, in the for-mulated delay-dependent queueing game, all gateways willcompete non-cooperatively but symmetrically to minimizetheir transmission costs in selecting β̂`,∀` ∈ L\{0}, for allnon-emergent medical packets. The formulated game can berepresented as

G , {N,B, {C`}L\{0}}, (16)

where all N gateways act as players, B denotes the sym-metric strategy set, and C` is the cost of gateways fortransmitting any packet in class `,∀` ∈ L\{0}. The Nashequilibrium (NE) of game G can be defined as follows.

Definition 2 (NE). A mixed strategy profile β̂e is a Nashequilibrium of game G if for any medical packet in class`,∀` ∈ L\{0}, β̂e` is a best response to β̂e in minimizingthe cost of gateways for transmitting this packet.

4.2 Waiting Delay Analysis

From the cost function (15), we can directly observe thatthe equilibrium of game G highly depends on the resultingdelay performance. Thus, we first provide some importantlemmas that depict the relationship between criticality coef-ficients, i.e., β`,∀` ∈ L\{0}, and mean waiting delays, i.e.,E[W (·)], in this prioritized queueing system.

Lemma 1. Let β̂ = {β̂`,∀` ∈ L\{0}} denote the mixedstrategy profile for all non-emergent medical packets.Then, the mean waiting delay for any packet in class`,∀` ∈ L\{0}, with a criticality coefficient β` can beexpressed in a recursive form as

E[W(β`,β̂)]=

φ1−ρ−

∑L`′=1

∫ β`

0 ρ`′E[W(β,β̂)](1− ββ`

)d(β̂`′(β))

1−∑L`′=1

∫∞β`ρ`′(1− β`

β )d(β̂`′(β)),

(17)where

ρ` = Λ`E[T`], ρ =L∑

`=1

ρ`, and φ =L∑

`=1

ρ`E[T 2` ]

2E[T`].

Proof: In the queueing system for delay-dependentprioritized beyond-WBAN transmissions, the waiting delayfor a non-emergent medical packet with criticality coeffi-cient β` consists of three parts, i.e., the remaining servicetime for packets that are currently under transmission, thetotal service time for future arrived packets that will over-take the considered packet in transmission, and the totalservice time for packets that are already waiting in the bufferand will be transmitted in prior to the considered packet.Thus, E[W (β`, β̂)] can be written as

E[W (β`, β̂)]=φ+E

L∑

`′=1

Y`′ (β`)∑

j=1

T`′(j) +

Z`′ (β`)∑

j=1

T`′(j)

,

(18)where φ denotes the expected remaining service time forpackets in transmission, T`′(j) is the service time of thepacket in class `′ with an index j, Y`′(β`) represents theexpected number of packets in class `′ that will arrive laterbut can be transmitted earlier, Z`′(β`) indicates the expectednumber of packets in class `′ that arrived earlier and will

also be transmitted earlier. Due to the independence be-tween the service time of the system and the arrival ofpacket transmission requests, (18) can be rewritten as

E[W (β`, β̂)] = φ+L∑

`′=1

E[T`′ ](Y`′(β`) + Z`′(β`)). (19)

To obtain Y`′(β`), let us consider that the medical packetwith β` has arrived at time 0 and waited for w. Then, anyother packets with β > β` that will arrive at time t > 0 suchthat β(w − t) > β`w can overtake the considered medicalpacket in delay-dependent prioritized beyond-WBAN trans-missions. In other words, all new arrived packets within(0, w(1 − β`

β )) will obtain higher transmission prioritiesthan the considered packet if they have β > β`. By thesuperposition and splitting property [43] of Poisson processand with some mathematical manipulations, we have

Y`′(β`) = Λ`′E[W (β`, β̂)]

∫ ∞

β`

(1− β`β

)d(β̂`′(β)), (20)

which calculates the expected number of packets in anyclass `′ with criticality coefficient larger than β` that arrivelater than the considered packet.

We are now left with computing Z`′(β`), i.e., the ex-pected number of packets that are already waiting in thebuffer and will not be overtaken by the considered medicalpacket with β`. Clearly, any buffered packet with β ≥ β` willdefinitely be transmitted before the considered one, and theexpected number of these packets can be calculated by

Z`′(β`)(1) = Λ`′

∫ ∞

β`

E[W (β, β̂)]d(β̂`′(β)). (21)

Besides, for any buffered packet with β < β` that arrived att− τ , the probability of not being overtaken by the medicalpacket with β` which arrived at t is equivalent to

Pr.(τ < E[W (β, β̂)] <β`

β` − βτ). (22)

Again, by the superposition and splitting property, we have

Z`′(β`)(2)

= Λ`′

∫ β`

0

[Pr.(τ < E[W (β, β̂)] <

β`β` − β

τ)

]dτd(β̂`′(β))

= Λ`′

∫ β`

0

[E[W (β, β̂)]− (1− β

β`)E[W (β, β̂)]

]d(β̂`′(β))

= Λ`′

∫ β`

0E[W (β, β̂)]

β

β`d(β̂`′(β)). (23)

which calculates the expected number of packets with crit-icality coefficient smaller than β` but will not be overtakenby the considered packet due to the sufficiently long experi-enced delays. In summary,

Z`′(β`) = Z`′(β`)(1) + Z`′(β`)

(2). (24)

Substituting (20) and (24) into (19) yields

E[W (β`, β̂)] = φ+L∑

`′=1

ρ`′

[∫ β`

0E[W (β, β̂)]

β

β`d(β̂`′(β))

+

∫ ∞

β`

(E[W (β`, β̂)](1− β`

β)+E[W (β, β̂)]

)d(β̂`′(β))

]. (25)

8

According to [44], φ can be well approximated by∑L`=1

ρ`E[T 2` ]

2E[T`] . Based on the conservation law [45], we have

L∑

`′=1

ρ`′

∫ ∞

β`

E[W (β, β̂)]d(β̂`′(β))

=φρ

1− ρ−

L∑

`′=1

∫ β`

0E[W (β, β̂)]d(β̂`′(β)).

(26)

Finally, substituting (26) into (25) along with some math-ematical manipulations, (17) can be derived.Lemma 2. For any non-emergent medical packet in class

`,∀` ∈ L\{0}, with a declared β`, we have(a) E[W (β`, β̂)] and d

dβ`E[W (β`, β̂)] are both continuous

with β`;(b) E[W (β`, β̂)] is a decreasing and convex function of β`.

Proof: To simplify the expression, we can rewrite (17)in Lemma 1 as

E[W (β`, β̂)] =

11−ρφ−

∑L`′=1 ρ`′H`′(β`)

1−∑L`′=1 ρ`′J`′(β`)

, (27)

where

H`′(β`) =

∫ β`

0E[W (β, β̂)]

(1− β

β`

)d(β̂`′(β)), (28)

J`′(β`) =

∫ ∞

β`

(1− β`

β

)d(β̂`′(β)). (29)

Since H`′(β`) and J`′(β`) are both continuous with β`,we can first conclude that E[W (β`, β̂)] is continuous withβ`. Now, let us further define H(β`) =

∑L`′=1 ρ`′H`′(β`),

and J (β`) =∑L`′=1 ρ`′J`′(β`). Then, the first order deriva-

tive of E[W (β`, β̂)] can be calculated as

d

dβ`E[W (β`, β̂)]

=−H′(β`)(1− J (β`)) + J ′(β`)

(φ

1−ρ −H(β`))

(1− J (β`))2

=E[W (β`, β̂)]J ′(β`)−H′(β`)

1− J (β`). (30)

The denominator of (30) is continuous because J (β`) iscontinuous. Thus, d

dβ`E[W (β`, β̂)] can be proved to be con-

tinuous with β` if the numerator of (30), i.e.,

F(β`) = E[W (β`, β̂)]J ′(β`)−H′(β`), (31)

is also continuous with β`. By expanding the expressions ofJ (β`) and H(β`), we have

F(β`) =L∑

`′=1

ρ`′

[E[W (β`, β̂)]

d

dβ`

∫ ∞

β`

(1− β`β

)d(β̂`′(β))

− d

dβ`

∫ β`

0E[W (β`, β̂)](1− β

β`)d(β̂`′(β))

]

= −L∑

`′=1

ρ`′

[∫ ∞

β`

E[W (β`, β̂)]1

βd(β̂`′(β))

+

∫ β`

0E[W (β`, β̂)]

β

β2`

d(β̂`′(β))

]. (32)

Since F(β`) shown in (32) is indeed continuous with β`,Lemma 2(a) holds.

Besides, F(β`) < 0 (which can be directly observed from(32)) together with J (β`) < 1 (which can be examined from(29)) indicates that

d

dβ`E[W (β`, β̂)] < 0. (33)

Moreover, it can be verified that J ′(β`) < 0, F ′(β`) > 0,and therefore

d2

dβ2`

E[W (β`, β̂)]=F ′(β`)(1−J (β`))+F(β`)J ′(β`)

(1− J (β`))2>0.

(34)Hence, a conclusion can be drawn from (33) and (34) thatE[W (β`, β̂)] is decreasing and convex with respect to β`.

4.3 Existence of Pure NEWith the help of Lemmas 1 and 2, in this subsection, weshow that there exists at least one pure NE in the formulateddelay-dependent prioritized queueing game G.Proposition 1. When declaring a beyond-WBAN transmis-

sion request, all gateways can follow a pure strategyprofile β = {β1, β2, . . . , βL}.

Proof: According to Lemma 2(b), the waiting delayof any non-emergent medical packet in beyond-WBANtransmissions is decreasing and convex with its criticalitycoefficient. This implies that the cost function C`(β`, β̂)given by (15) is a convex and continuous function of β`. Inorder words, there is a unique best response in determiningβ` with the objective of minimizing C`(β`, β̂) regardless ofβ̂. Thus, it is without loss of generality to consider a purestrategy profile β = {β1, β2, . . . , βL} in game G.Theorem 1. The formulated delay-dependent prioritized

queueing game G has at least one pure NE.

Proof: By definition in (15), the cost function fortransmitting each non-emergent medical packet under thepure strategy profile β can be expressed as

C`(β`,β) = θ`E[W (β`,β)] + β`, ∀` ∈ L\{0}. (35)

Since it has already been proved that C`(β`,β) is convexand continuous with respect to β`, in order to minimizeC`(β`,β), it is intuitive that β` will be determined with thesatisfaction of

C`(β`,β) ≤ C`(0,β), ∀` ∈ L\{0}. (36)

Substituting (35) into (36), we have

θ`E[W (β`,β)] + β` ≤ θ`E[W (0,β)], (37)

which is equivalent to

β` ≤ θ`E[W (0,β)] = θ` ·φ

(1− ρ)2= β̄`, (38)

where E[W (0,β)] = φ/(1 − ρ)2 is obtained by substitutingβ` = 0 into (17) as

E[W (0,β)]=

φ1−ρ−

∑{`′|β`′<0} ρ`′(1−

β`′β`

)E[W (β`′ ,β)]

1−∑{`′|β`′≥0} ρ`′(1− 0

β`′)

=φ

(1− ρ)2. (39)

9

Then, we can limit the strategies in declaring packets’ crit-icality coefficients to a compact and convex set in the formof Cartesian product, i.e.,

B =L∏

`=1

[0, β̄`] ⊂ RL. (40)

Since the strategy set B is nonempty, convex and com-pact, and the cost function C`,∀` ∈ L\{0}, is continuousand convex, the game G has at least one pure NE.

4.4 Properties and Computation of NE

Replacing the mixed strategy profile β̂ by the pure strategyprofile β in (17), the mean waiting delay for any non-emergent medical packet with a criticality coefficient β` canbe rewritten as

E[W (β`,β)]=

φ1−ρ−

∑{`′|β`′<β`} ρ`′(1−

β`′β`

)E[W (β`′ ,β)]

1−∑{`′|β`′≥β`} ρ`′(1−

β`

β`′)

.

(41)Note that there is a recursive term in the numerator of (41),so that the cost function, i.e., C`(β`,β),∀` ∈ L\{0}, cannotbe explicitly expressed. This implies that it is impossible toanalyze the NE of the delay-dependent prioritized queue-ing game G through traditional ways (in non-cooperativegame theory), such as checking the standard conditionsor verifying the diagonal concavity of the best-responsefunctions [46]. Hence, to avoid these inherent difficulties,in the following, we will explore the properties of thecorresponding NE with a joint consideration of the beyond-WBAN queueing performance.

Lemma 3. Let βe = (βe1 , βe2 , . . . , β

eL) be the NE of queueing

game G. If θ1 ≥ θ2 ≥ . . . ≥ θL, then we should have

βe1 ≥ βe2 ≥ . . . ≥ βeL. (42)

Proof: Assume in the way of contradiction that

βe` > βe`′ , ∃` > `′. (43)

By the definition of NE, we have

θ`E[W (βe` ,βe)] + βe` ≤ θ`E[W (βe`′ ,β

e)] + βe`′ , (44)

which is equivalent to

βe` − βe`′ ≤ θ`(E[W (βe`′ ,βe)]− E[W (βe` ,β

e)]). (45)

Since θ` ≤ θ`′ , we have

βe` − βe`′ ≤ θ`′(E[W (βe`′ ,βe)]− E[W (βe` ,β

e)]). (46)

Clearly, (46) can also be rewritten as

θ`′E[W (βe`′ ,βe)] + βe`′ ≥ θ`′E[W (βe` ,β

e)] + βe` , (47)

which obviously contradicts the assumption in (43) that βe`′is the NE for medical packets in class `′.

Lemma 3 indicates an intuition that medical packetswith higher delay sensitivities will always be reported withhigher criticality coefficients which reflect their medical im-portance in the delay-dependent prioritized beyond-WBANtransmission scheduling system.

Lemma 4. The NE of game G, i.e., βe = (βe1 , βe2 , . . . , β

eL),

satisfies the condition that

E[W (βe` ,βe)] =W(βe` ,β

e), ∀` ∈ L\{0}, (48)

where

W(β`,β)

=1−∑``′=1ρ`′(1−

β`

β`′)

θ`∑``′=1

ρ`′β`′

−∑L`′=`+1ρ`′β`′E[W (β`′ ,β)]

β2`

∑``′=1

ρ`′β`′

.

(49)

Proof: Taking the derivative of (41) and consideringthe order obtained in Lemma 3, we have

dE[W (β`,β)]

dβ`

=

∑``′=1

ρ`′β`′

( φ1−ρ −

∑L`′=`+1 ρ`′(1−

β`′β`

)E[W (β`′ ,β)])

(1−

∑``′=1 ρ`′(1−

β`

β`′))2

−

∑L`′=`+1 ρ`′

β`′β2`

(1−

∑``′=1 ρ`′(1−

β`

β`′))

(1−

∑``′=1 ρ`′(1−

β`

β`′))2 . (50)

The best response of the criticality coefficient selectionfor each medical packet in class `,∀` ∈ L\{0}, is given by

dC`(β`,β)

dβ`= θ`

dE[W (β`,β)]

dβ`+ 1 = 0. (51)

Then, substituting (50) into (51) yields

1−∑̀

`′=1

ρ`′(

1− β`β`′

)= θ`

( 1

β2`

L∑

`′=`+1

ρ`′β`′E[W (β`′ ,β)]

+ E[W (β`,β)]∑̀

`′=1

ρ`′

β`′

).

(52)

After some mathematical manipulations on (52), we caneventually derive the condition shown in (48) and (49).

Notice that both E[W (β`,β)] andW(β`,β) are recursiveformulae in terms of the mean waiting delays, and thusthe NE of game G has to satisfy these two recursionssimultaneously.

Proposition 2. Given a strategy profile β in the order of (42),W(β`,β),∀` ∈ L\{0}, has the following properties:

(a) W(β`,β) is increasing with β` ∈ (β`+1, β`−1);(b) W(β`,β) is decreasing with β`′∈ (β`′+1, β`′−1),∀`′ > `.

Proof: From the expression ofW(β`,β) in (49), we canobserve thatW(β`,β) monotonically increases with β` if

d

dβ`

1

β2`

E[W (β`′ ,β)] ≤ 0, ∀`′ > `. (53)

From (41), we have

E[W (β`′ ,β)]

β2`

=

φ1−ρ−

∑L`′′=`′+1 ρ`′′(1−

β`′′β`′

)E[W (β`′′ ,β)]

β2` (1−

∑`′

`′′=1 ρ`′′(1−β`′β`′′

)),

(54)which is apparently decreasing with β`, and hence Proposi-tion 2(a) holds.

10

Next, we will show that W(β`,β) is a decreasing func-tion of β`′ ,∀`′ > `. Taking the derivative of W(β`,β) withrespect to β`′ ,∀`′ > `, we have

d

dβ`′W(β`,β)

∝−ρ`′dβ`′E[W (β`′ ,β)]

dβ`′−∑

{`′′ 6=`′,`′′>`}

ρ`′′β`′′dE[W (β`′′ ,β)]

dβ`′,

(55)

where the symbol “∝” stands for “be proportional to”. Sinced

dβ`′E[W (β`′′ ,β)] ≥ 0,∀`′′ 6= `′, the summation term in (55)

is non-negative. Moreover, it is not difficult to verify thatβ`′E[W (β`′ ,β)] is increased with β`′ . Thus, d

dβ`′W(β`,β) ≤

0 and Proposition 2(b) is proved.With Proposition 2 and Lemmas 2, 3 and 4, we can obtain

the following corollary.Corollary 1. For each non-emergent medical packet in class

`,∀` ∈ L\{0}, there is at most one β` ∈ (β`+1, β`−1)which can satisfy the equilibrium condition (48).

Even though we are not able to analytically express theNE, i.e., βe = {βe1 , βe2 , . . . , βeL}, of the delay-dependentprioritized queueing game G, different numerical methods[47], [48] can be applied to compute the NE, and the resultswill be demonstrated in Section 5. For a special case that allnon-emergent medical packets are equally sensitive to thepotential waiting delays, a closed-form NE can be derived.Theorem 2. If θ1 = θ2 = . . . = θL = θ̃, then the unique NE

of the criticality coefficients is

βe` =θ̃φρ

1− ρ, ∀` ∈ L\{0}. (56)

Proof: Please see Appendix A.

To further measure the performance of game G on mini-mizing the social cost (i.e., mean waiting cost of all medicalpacket transmissions in the beyond-WBAN), we introducethe definition of price of anarchy (PoA) as follows:

PoA =Cost(NE)

Cost(OPT )=

∑L`=1 Λ`θ`E[W (βe` ,β

e)]∑L`=1 Λ`θ`E[W ζ∗

` ], (57)

where Cost(NE) and Cost(OPT ) denote the social costsresulted by the NE βe of game G and the centralized optimalscheduling scheme ζ∗, respectively. Obviously, we musthave PoA > 1. The detailed derivation of ζ∗ is shown inAppendix B, and the corresponding numerical study of PoAwill be discussed in Section 5.

5 SIMULATION RESULTS

We use MATLAB as the simulation tool to examine our the-oretical analyses and evaluate performance of the proposedmanagement framework for delay-sensitive medical packettransmissions in the beyond-WBAN. All results are obtainedby using Monte Carlo simulation to build up the queueingmodel and run the noncooperative game.

5.1 Simulation SettingsConsider a beyond-WBAN with one BS who owns K =10 channels for scheduling both emergent and non-emergent medical packet transmissions. As defined in IEEE

Probability of emergent alarms0.05 0.1 0.15 0.2 0.250

2

4

6

8

10

QoS threshold (seconds)0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Nu

mb

er

of

ch

an

ne

ls

0

2

4

6

8

10

K0

min for emergency

K-K0

min for non-emergency

K-K0

min for non-emergency

K0

min for emergency

Fig. 3. Channel assignments for emergency and non-emergency.

802.15.6 [27], all medical packets are categorized into L ={0, 1, . . . , 7} classes, where {0} and {1, 2, . . . , 7} denotethe sets of classes for emergent alarms and non-emergentroutines, respectively. According to medical data character-istics [49], the size of each medical packet is chosen in therange of [50, 100] Kb, and the QoS threshold (i.e., delayrequirement) for emergent information delivery is set as200 ms. The beyond-WBAN transmission rate is assumedto be uniformly distributed over [250, 500] Kbps, whichis identical to the uplink transmission rate in 3G cellularnetworks [50]. The average arrival rate of medical packetsin each class `,∀` ∈ L, is Λ` per second, where Λ` ischosen as an integer from 1 to 6. Furthermore, define a totalarrival rate of non-emergent medical packet as Λ, in whicheach class of packets has the same arrival rate. Besides, letdelay sensitivity of non-emergent packets, θ1, θ2, . . . , θ7, be0.7, 0.6, . . . , 0.1, respectively. Similar settings have also beenemployed in [4], [6]. In the following, some parameters mayvary depending on different simulation scenarios.

5.2 Performance Evaluation

Fig. 3 shows the calculation results of channel assignmentfor emergent and non-emergent medical packet transmis-sions in the considered beyond-WBAN. At least Kmin

0

channels have to be reserved to guarantee the mQoS ofemergent alarms. It can be seen that Kmin

0 decreases withthe relaxation of the emergent packets’ delay requirement(i.e., QoS threshold), while increases with the probabilityof emergent alarms. As an example, we can see that, inthe scenario that the probability of emergent alarms is lessthan 0.1 and the QoS threshold is 200 ms [49], Kmin

0 = 2channels are enough to meet the requirement of emergentinformation delivery, and the rest ofK−Kmin

0 = 8 channelscan be assigned for non-emergent packet transmissions.Note that calculation results obtained in Fig. 3 will be usedfor follow-up simulations.

As mentioned in Section IV-A, the employment of sym-metric strategy profile relies on the approximation that allgateways face the same expected waiting delay (and thusthe same cost function) for transmitting medical packetsbelonging to the same class `,∀` ∈ L\0. Specifically, for

11

Number of gateways20 30 40 50 60 70 80 90 100

De

lay d

iffe

ren

ce

of

pa

cke

t tr

an

sm

issio

n (

se

co

nd

s)

0

0.5

1

1.5

2

2.5

Number of channels K=5

Number of channels K=10

Number of channels K=20

Fig. 4. Difference of waiting delay performance between scenarios withsymmetric and asymmetric strategies.

medical packets belonging to the same class, the cost func-tion (14) can be rewritten as

Cn = θE[W (βn)] + π(βn),

where n denotes the index of a gateway. By consider-ing symmetric strategies among gateways, we imply thatall gateways face the same expected waiting delay, i.e.,E[W (βn)], in the system. Although, in practice, E[W (βn)]may be different among gateways due to different channelqualities so that strategies tend to be asymmetric, it isobvious from the queueing perspective that E[W (βn)] ismostly affected by the aggregate performance of all gate-ways’ strategies and their channel qualities rather thaneach gateway itself, especially when the total number ofgateways is relatively large, so that such heterogeneity maybe negligible. In order to evaluate the performance of thissymmetric approximation, we show in Fig. 4 the differ-ence of waiting delays between scenarios with symmetricand asymmetric strategies with respect to the number ofgateways, where the waiting delay in symmetric strategyscenario is calculated by assuming that the channel qualitiesof all gateways follow the same distribution, and the onein asymmetric strategy scenario is obtained by consideringarbitrarily different channel qualities of different gateways.From this figure, we can see that the delay difference de-creases quickly with the increase of the number of gateways.Besides, it shows that more channels (i.e., a larger K) leadto even smaller delay difference. This is because the impactfrom an individual gateway on the stability of the overallqueueing system becomes smaller when the scale of thesystem (i.e., the number of gateways and channels) is larger.Moreover, since the delay difference approaches 0 when thenumber of gateways equals 100 (which is a common settingin cellular networks), we can conclude that the consider-ation of symmetric strategy profile has a relatively goodapproximation performance in practical beyond-WBANs.

Fig. 5 illustrates the maximum values (or the ranges)of criticality coefficients (strategies), i.e., β̄`, for each non-emergent medical packet in class `,∀` ∈ L\{0}. From thisfigure, we can see that β̄` increases with the packet arrivalrate Λ. This is because the competition among gateways inbeyond-WBAN transmissions becomes much more intense

Total arrival rate of non-emergent medical packets10 15 20 25 30 35 40

Ma

xim

um

va

lue

of

critica

lity c

oe

ffic

en

t β

0

0.5

1

1.5

2

2.5

θ1=0.7

θ4=0.4

θ7=0.1

Fig. 5. β̄` for non-emergent medical packets in class `.

Criticality coefficient0.5 1 1.5 2 2.5

Wa

itin

g d

ela

y (

se

co

nd

s)

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

Total arrival rate of non-emergent packets Λ=42

Total arrival rate of non-emergent packets Λ=28

Total arrival rate of non-emergent packets Λ=14

Fig. 6. Waiting delay vs. different criticality coefficients.

when the queueing system is busier. More specifically, alarger Λ indicates longer waiting delays, especially forpackets with smaller criticality coefficients, so that eachintelligent gateway may strategically determine higher crit-icality coefficients for their medical packets in order to po-tentially obtain better beyond-WBAN transmission servicesand lower transmission costs. Due to the non-cooperativenature among gateways, this would result in a more intensecompetition in strategically increasing β`, and eventuallylead to the increment of β̄`.

In Fig. 6, the mean waiting delays of a randomly selectednon-emergent medical packet with heterogeneous criticalitycoefficients are investigated. It is shown that the largercriticality coefficient the packet has, the shorter mean wait-ing delay it obtains. This is because more critical medicalpackets will be transmitted in higher priorities than thosewith less emergency. By further comparing the curves withdifferent total packet arrival rate Λ, we can see that thewaiting delay is longer when Λ increases. It is intuitivethat a larger Λ implies that more packets are required tobe transmitted so that the waiting probability increases.

Fig. 7 shows the expected costs for transmitting differ-ent classes of non-emergent medical packets with hetero-geneous reported criticality coefficients. The trend of thecurves in this figure indicates that the transmission costof each packet in class `,∀` ∈ L, first decreases with thecriticality coefficient β`. This is because, with the increase

12

Criticality coefficient0.5 1 1.5 2 2.5

Tra

nsm

issio

n c

ost

of

the

pa

cke

t

2.2

2.4

2.6

2.8

3

3.2

3.4

θ1=0.7

θ2=0.6

θ3=0.5

Fig. 7. Expected transmission cost vs. different criticality coefficients.

Criticality coefficient0.5 1 1.5 2 2.5

E[W

(βl,β

)]

0

1

2

3

4

5

6

7

0.5 1 1.5 2 2.5

W(β

l,β)

0

1

2

3

4

5

6

7

Waiting delay for packets in class 6

Waiting delay for packets in class 2

NE condition for packets in class 6

NE condition for packets in class 2

NE

β6

eβ

2

e

Fig. 8. NE analysis of the delay-dependent prioritized queueing game.

of β`, a shorter waiting delay is granted for the packettransmission so that the waiting cost is reduced. However,after a certain point, since the delay requirement has alreadybeen met, the service charge becomes dominant, and thetotal transmission cost increases. This numerically provesthat the packet’s transmission cost is convex with its critical-ity coefficient, and thus Proposition 1 holds. Furthermore,given a same criticality coefficient, it is intuitive that thepacket with a higher delay sensitivity suffers more waitingcost, which results in a larger expected transmission cost.

The properties of NE for the formulated delay-dependent prioritized queueing game G is examined inFig. 8. Here, the NE is derived by the numerical methodknown as best response dynamics [48]. The idea is to itera-tively update the strategy of each player in the game byan arbitrarily beneficial deviation until the NE has beenreached [51]. In our considered scenario, we can initially setcriticality coefficients β` = 0,∀` ∈ {1, . . . , 7}, and calculatethe transmission costs, i.e., C`(β`,β), for medical packets ineach class `. Then, starting from class ` = 1 (with the highestdelay sensitivity as defined in (13)), we can seek the bestresponse by gradually increasing β1 while fixing the valuesof β\β1, until C1 is minimized. This adaption procedurecontinues sequentially from class ` = 1 to 7. After that, weiteratively check the best responses by following the afore-mentioned order, and stop the iterations when no furtheradaption is required. By fixing the NE of all other packets

TABLE 2Analysis on PoA of the formulated queueing game.

Strategy update order ` = 1 to 7 ` = 7 to 1Strategy update rate ∆ = 0.01 ∆ = 0.1 ∆ = 0.01 ∆ = 0.1

PoA Λ = 14 1.29 1.33 1.47 1.53Λ = 42 1.58 1.71 1.89 2.03

and adjusting the strategy of one packet in any class `,we can observe that E[W (β`,β)] decreases, whileW(β`,β)increases with β`. Moreover, there is a unique intersectionof these two curves such that E[W (β`,β)] = W(β`,β), andthe corresponding criticality coefficient is exactly the NE ofthis packet, i.e., βe` . In addition, it is also shown that the NEof medical packets with a higher delay sensitivity is largerthan that with a lower delay sensitivity (e.g., βe2 > βe6). Thus,Lemmas 3, 4 and Corollary 1 are verified.

In Table 2, the PoA of the formulated queueing game isnumerically evaluated with respect to two different strategyupdate orders (i.e., starting from class ` = 1 to 7 and fromclass ` = 7 to 1) and two different strategy update rates (i.e.,the gradual increment on criticality coefficient ∆ = 0.01and ∆ = 0.1). According to the definition in (57), a smallervalue of PoA indicates a better performance on minimizingthe social cost. From this table, we can see that the PoAobtained by updating strategy from class ` = 1 (withthe highest delay sensitivity) to 7 (with the lowest delaysensitivity) is better than that from ` = 7 to 1. This is becausegiving higher strategy update priorities to medical packetswith higher delay sensitivities can always result in a moreappropriate transmission scheduling (i.e., shorter expecteddelays for packets with higher delay sensitivities) and thuslower the social waiting cost. Besides, it is shown that asmaller ∆ always leads to a better performance (i.e., a lowerPoA). This is because a smaller ∆ implies a larger searchingspace in finding the NE. In addition, it is intuitive that theresulted PoA increases with the total packet arrival rate Λ.Thus, we can conclude from this table that the proposedqueueing game approach has a good overall performancein minimizing the social cost if the strategy update order isfrom ` = 1 to 7 and the update rate is 0.01, especially forunderloaded systems.

Fig. 9 reveals the relationship between the service chargeof packets with different delay sensitivities and the totalarrival rate of non-emergent medical packets. It can be seenfrom the figure that the service charges for packets withhigher delay sensitivities are always larger. This matchesthe intuition that the packet with a more stringent QoSrequirement so as to receive a better service (i.e., a shorterdelay) has to pay more for its beyond-WBAN transmission.Besides, since the competition in the queueing game be-comes more and more intense when the system is gettingincreasingly busier as explained in Fig. 5, the service chargesin Fig. 9 also increases with the packet arrival rate.

5.3 Comparisons with Existing Scheduling SchemesIn this subsection, the superiority of the proposed beyond-WBAN management framework is demonstrated. For com-parison purpose, the transmission scheduling schemesbased on absolute priority rule [4], FCFS discipline [24] and

13

Total arrival rate of non-emergent medcial packets10 15 20 25 30 35 40

Se

rvic

e c

ha

rge

0

0.2

0.4

0.6

0.8

1

1.2

Medical packets in class 1

Medical packets in class 4

Medical packets in class 7

Fig. 9. Charge for each non-emergent medical packet transmission.

Total arrival rate of non-emergent medical packets10 15 20 25 30 35 40

Va

ria

nce

of

pa

cke

ts' w

aitin

g d

ela

ys

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

5.5

Transmission scheduling based on absolute priority

Transmission scheduling based on FCFS

Transmission scheduling based on PF

Proposed delay-dependent prioritized scheduling

Fig. 10. Comparison on the variance of medical packets’ waiting delays.

proportional fairness (PF) [52] are simulated as benchmarks.The absolute priority rule maintains a fixed transmissionorder regardless of packets’ experienced waiting delays, theFCFS discipline treats all packet transmissions equally with-out the consideration of mQoS requirements, and the PFscheduling aims to maximize the overall network through-put while ensuring that each gateway can be allocated withat least a minimum amount of transmission opportunitiesfor balancing the tradeoffs between efficiency and fairness.

Fig. 10 shows the difference between mean waiting de-lays for all classes of medical packets (which is calculatedby variance) under different scheduling schemes. It can beobserved that the scheduling scheme based on the absolutepriority rule produces a considerably large variance on wait-ing delays because its beyond-WBAN transmission servicesfor less important medical packets can be tremendouslyworse than critical ones. In contrast, scheduling schemesbased on both the FCFS and PF can fairly grant all medicalpackets with the same transmission opportunities, and thusresults in zero variance. Our propose scheduling schemejointly takes into account the original medical packet crit-icality and packets’ experienced waiting delays, so that itleads to a relatively low variance of packets’ mean waitingdelays. More importantly, unlike scheduling schemes basedon FCFS and PF, the proposed scheme can also guaranteethe mQoS requirement for medical information delivery,and such superiority will be further illustrated in Fig. 11.

Total arrival rate of non-emergent medical packets10 15 20 25 30 35 40

Me

an

wa

itin

g c

ost

of

all

me

dic

al p

acke

ts

0

0.5

1

1.5

2

2.5

3

3.5

4

Transmission scheduling based on absolute priority

Transmission scheduling based on FCFS

Transmission scheduling based on PF

Proposed delay-dependent prioritized scheduling

Fig. 11. Comparison on all packets’ waiting cost in the beyond-WBAN.

Fig. 11 compares different scheduling schemes in termsof the mean waiting cost of all medical packets in beyond-WBAN transmissions. Since medical packets suffer higherwaiting delays in a busier queueing system as illustrated inFig. 6, it is intuitive that the waiting cost increases with thepacket arrival rate. Besides, the FCFS discipline results in thehighest waiting cost due to ignoring the potential prioritiesamong different packet classes. The scheduling schemesbased on PF and absolute priority can achieve better perfor-mance than FCFS because of the improvement on networkthroughput and the consideration of medical-grade priority,respectively. Moreover, it is shown that the proposed delay-dependent prioritized scheduling scheme outperforms allthe others. This is because the proposed scheme well balancewaiting costs of emergent and non-emergent packets byproviding the most appropriate transmission services basedon both packet criticality and experienced waiting delays.

6 CONCLUSION

In this paper, we study a novel management frameworkfor beyond-WBAN transmissions of medical packets withheterogeneous delay sensitivities. The network model withone BS and multiple smart WBAN-gateways is considered.By taking into account the random arrivals of medicalpackets and the potential strategic behaviors of gateways,a non-cooperative delay-dependent prioritized queueinggame is formulated. A new theoretical framework is pro-posed to jointly characterize the queueing performances andthe properties of the game equilibrium. Simulation resultsdemonstrate that our proposed transmission schedulingscheme is feasible and applicable in beyond-WBANs.

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APPENDIX APROOF OF THEOREM 2

Proof: Consider that the criticality coefficient for anon-emergent medical packet in class ` is strategically de-termined as β`, while the criticality coefficients for all otherclasses, i.e., L\{0, `}, are the same, denoted by β.

Based on (41), the mean waiting delay for packets in class` can be calculated by considering the following 3 cases:

• If β` > β,

E[W (β`, β)] =

φ1−ρ − ρ(1− β

β`)E[W (β, β)]

1− 0

=φ

1− ρ· (1− ρ)β` + ρβ

β`;

(58)

• If β` = β,

E[W (β`, β)] =

φ1−ρ − 0

1− 0=

φ

1− ρ; (59)

• If β` < β,

E[W (β`, β)] =

φ1−ρ − 0

1− ρ(1− β`

β )=

φ

1− ρ· β

β(1− ρ) + ρβ`.

(60)

Since the second case can be considered as a special casein either the first or third one,

E[W (β`, β)] =

φβ`(1− ρ) + φβρ

β`(1− ρ), β` ≥ β;

φβ

β(1− ρ)2 + β`ρ(1− ρ), β` < β.

(61)

Taking the derivative on (61) with respect to β`, we have

dE[W (β`, β)]

dβ`=

−φβρ

(1− ρ)β2`

, β` ≥ β;

−φβρ

(β(1−ρ)+β`ρ)2(1−ρ), β` < β.

(62)

Furthermore, we can easily verify that

limβ`→β+

dE[W (β`, β)]

dβ`= limβ`→β−

dE[W (β`, β)]

dβ`=−φρ

β(1− ρ). (63)

Then, this theorem can finally be proved by substituting(63) into (51).

APPENDIX BDERIVATION OF OPTIMAL SCHEDULING SCHEME ζ∗

The optimal scheme ζ∗ can be obtained by solving a purequeueing scheduling problem with the aim of minimizing

the social cost as

[ζ∗] = arg minL∑

`=1

Λ`θ`E[W ζ` ] (64)

s.t., (E[W ζ1 ],E[W ζ

2 ], . . . ,E[W ζL ]) ∈ Q(ζ), (65)

where Q(ζ) represents the queueing system depending onthe scheduling scheme ζ. Inspired by [53], the optimalscheduling scheme ζ∗ can be designed according to thefollowing theorem.Theorem 3. Given a set of {1, 2, . . . , L} classes of medical

packets, reindex them based on a decreasing order ofθ`/E[T`],∀` ∈ {1, 2, . . . , L}, i.e.,

θ1

E[T1]≥ θ2

E[T2]≥ . . . ≥ θL

E[TL]. (66)

Then, to minimize the social cost, ζ∗ will always grant ahigher transmission priority to medical packets in class` than those in class ` + 1,∀` ∈ {1, . . . , L − 1} with thenew indices defined by (66).

Proof: This is equivalent to prove that for medicalpackets belonging to any two consecutive classes (withindices defined by (66)), e.g., ` and `+ 1, under the optimalscheduling scheme ζ∗, packets in class ` will always betransmitted in an absolutely higher priority over the onesin class `+ 1.

The expected waiting costs for medical packets in bothclasses ` and `+ 1 can be written as

Cost(`,`+1) =Λ`θ`E[W`] + Λ`+1θ`+1E[W`+1]

=θ`

E[T`]ρ`E[W`]+

θ`+1

E[T`+1]ρ`+1E[W`+1],

(67)

where ρ` = Λ`E[T`]. By the Little’s law, we have

ρ`E[W`] = Λ`E[T`]E[W`] = E[T`]E[Q`], (68)

where E[Q`] represents the expected queue length of the`-th class packets, and thus E[T`]E[Q`] denotes the meanworkload of the `-th class packets in the queue. Obvi-ously, the total workload of both the `-th and the ` + 1-thclass medical packets in the queue remains a constant, i.e.,E[T`]E[Q`] + E[T`+1]E[Q`+1] is a constant, regardless of ζ∗.

Therefore, if θ`/E[T`] ≥ θ`+1/E[T`+1], the only wayto minimize Cost(`,`+1) is to decrease E[W`] as much aspossible (even though E[W`+1] will increase accordingly).In other words, medical packets in class ` should be grantedby a higher priority over those in class `+ 1.

Theorem 3 indicates that for achieving social cost mini-mization, medical packets in a class with a higher cost perunit of waiting delay or a shorter expected service timeshould be given an absolutely higher transmission priority,regardless of experienced waiting delays.

16

Changyan Yi (S’16) received the B.Sc. degreefrom Guilin University of Electronic Technology,China, in 2012, and M.Sc. degree from Univer-sity of Manitoba, MB, Canada, in 2014. He is cur-rently working toward the Ph.D. degree in Elec-trical and Computer Engineering, University ofManitoba. He was awarded A. Keith Dixon Grad-uate Scholarship in Engineering for 2017-2018,Edward R. Toporeck Graduate Fellowship in En-gineering for 2014-2017 (four times), Universityof Manitoba Graduate Fellowship (UMGF) for

2015-2018, and IEEE ComSoc Student Travel Grant for IEEE Globecom2016. His research interests include algorithmic game theory, queueingtheory and their applications in radio resource management, wirelesstransmission scheduling and network economics.

Jun Cai (M’04-SM’14) received the B.Sc. andM.Sc. degrees from Xi’an Jiaotong University,Xi’an, China, in 1996 and 1999, respectively, andthe Ph.D. degree from the University of Waterloo,ON, Canada, in 2004, all in electrical engineer-ing. From June 2004 to April 2006, he was withMcMaster University, Hamilton, ON, as a NaturalSciences and Engineering Research Council ofCanada Postdoctoral Fellow. Since July 2006, hehas been with the Department of Electrical andComputer Engineering, University of Manitoba,

Winnipeg, MB, Canada, where he is currently an Associate Profes-sor. His current research interests include energy-efficient and greencommunications, dynamic spectrum management and cognitive radio,radio resource management in wireless communications networks, andperformance analysis. Dr. Cai served as the TPC Co-Chair for IEEEVTC-Fall 2012 Wireless Applications and Services Track, IEEE Globe-com 2010 Wireless Communications Symposium, and IWCMC 2008General Symposium; the Publicity Co-Chair for IWCMC in 2010, 2011,2013, and 2014; and the Registration Chair for QShine in 2005. Healso served on the editorial board of the Journal of Computer Systems,Networks, and Communications and as a Guest Editor of the specialissue of the Association for Computing Machinery Mobile Networksand Applications. He received the Best Paper Award from Chinacom in2013, the Rh Award for outstanding contributions to research in appliedsciences in 2012 from the University of Manitoba, and the OutstandingService Award from IEEE Globecom in 2010.