Coupled reservation protocols for hierarchical single- hop photonic networks

M.W. Janoska T. D. Todd

Indexing terms: Single-hop architectures, Passive star networks, Wavelength routing, Remote media access protocols

Abstract: Future photonic LANs and MANs may be based upon single-hop architectures using wavelength division multiplexing (WDM) and passive star coupling. Recently, however, single- hop designs have been introduced which use a hierarchical structure employing simple wavelength routing functions. A salient feature of this type of network is that a set of wavelengths can be reused locally, while maintaining full single-hop connectivity through a different set of globally shared wavelengths. In such a network, wavelength routing divides the system into separate local and remote subnetworks, which may be accessed from each station by a single tunable transmitter. As a result, media access protocol operation can be implemented independently for each subnetwork, or it may be coupled between levels. These two options are investigated, focusing on remote subnetwork performance. Since global connectivity is achieved using a single set of remote wavelengths, it is important that they are efficiently used. The proposed protocols implement different forms of dynamic movable boundary TDMA, using various request/allocation mechanisms motivated by the channel controller approach. It is shown that, in such networks, remote channel efficiency can be improved significantly through a strong coupling of local and remote system operation.

List of symbols

C, = local wavelength channels CR = remote wavelength channels L = data slot size relative to remote request packet

a = remote allocation size relative to remote request

A = LONs in the network N = stations on a local HOME channel N,, = total stations in the network

length

packet length

0 IEE, 1997 ZEE Proceedings online no. 19971349 Paper first received 30th January 1996 and in revised form 19th March 1997 The authors are with the Communications Research Laboratory, McMaster University, Hamilton, Ontario, Canada L8S 4K1

DR = data slots per remote channel DL = data slots per local channel MR = number of remote request/allocation minislots P = station-star propagation delay relative to remote

request packet length

1 Introduction

A promising architecture for future LANs may be con- structed using a passive optical star coupler combined with wavelength division multiplexing [l]. In such a system, however, if the number of available channels is small, then the scalability of the system may be very poor, even in a local area design. In this case, an improved system may be created using a two level hier- archy by employing simple wavelength routing func- tions [2-41. In this type of system, multiple subnetworks can spatially reuse a set of local wave- lengths, while a set of remote wavelengths is shared globally. Such a design can exploit the high degree of locality among user stations which is typical in many MANs and large LANs. In this case, stations belonging to the same community of interest would be configured as a separate subnetwork, if possible.

In a single-hop hierarchical optical network, wave- length routing creates separate local and remote sub- networks which may be accessed from each station. This is accomplished by using a wavelength agile source such as a tunable laser or transmitter array. In this case, media access protocol operation can be implemented independently for each subnetwork, or it may be coupled between levels. In this paper, we inves- tigate protocols for these two options, focusing on remote subnetwork performance. Since global connec- tivity is maintained by a single set of non-reused remote wavelengths, it is especially important that remote media access is performed as efficiently as pos- sible. By contrast, the performance of the local media access protocol may be less critical, since wavelengths are reused across multiple local subnetworks. The pro- tocols considered implement various forms of dynamic movable boundary TDMA, using request/allocation mechanisms implemented using channel controllers [5 ] . Simulation and analytical models are used to character- ise protocol delay and capacity performance. It is found that the most efficient schemes require a strong coupling of local and remote system operation. In these cases, remote channel efficiency may be significantly improved without adversely affecting local system per- formance.

IEE Proc.-Commun., Vol. 144, No. 4, August 199i 241

2 Network architecture and operation

The physical architecture of the network is an enhanced version of those discussed in [24]. The lim- ited scalability in a single passive star network is addressed through spatial wavelength reuse. Fig. 1 illustrates the basic physical topology. The network is divided into a hierarchy consisting of multiple local optical networks, or LONs, and a single remote optical network, or RON. At the lower level, the LONs oper- ate independently and concurrently. At the higher level, the remote optical network provides connectivity between the LONs. In the Figure, two LONs are shown connected to the RON at the top of the dia- gram. Additional details are shown in the LON on the left of the Figure.

The available wavelengths are divided into two con- tiguous sets, referred to as the local waveband, AL, and the remote waveband, AR. It is assumed that there are CL and C, channels in the respective local and remote wavebands. The local channels are used for intra-LON communication and are spatially reused in each LON. The remote channels are used for inter-LON communi- cation and are globally shared among all stations. The design is such that any station can access both wave- bands directly, using the wavelength agility of its trans- mitter. In addition to these basic functions, in this paper we include an innovation whereby certain con- trollers also reuse the local channels over the remote network for signalling purposes. This function is referred to as remote reuse signalling.

A 3 x 3 WDM cross connect links each LON into the RON and performs three wavelength routing func- tions, as shown in Fig. 1. First, it routes the local waveband from the local input fibre to the local output fibre, thus creating a broadcast-and-select topology within each LON using AL. The second function is to route the remote waveband from the local input fibre to the remote output fibre and the remote waveband from the remote input fibre to the local output fibre. Since all remote cross connect ports terminate at the

remote star coupler, this allows all remote wavelengths to be shared across all LONs. The final function is to create the remote reuse signalling function, discussed above, by routing the local waveband from the LMC input fibre to the remote output fibre and the local waveband from the remote input fibre to the LMC out- put fibre. This functionality is used by one of the remote media access protocols and will be described later.

In the network considered, station media access is motivated by the single star DCCN design described in [5]. DCCN uses a dynamic requestiallocation mecha- nism, where each wavelength division multiplexing (WDM) channel is assigned a channel controller (CC) which performs various functions such as scheduling and synchronisation. The motivation is to simplify the user stations as much as possible so that minimal hard- ware realisations can be obtained. In the hierarchical network we adopt a similar approach. Accordingly, in the basic design, a channel controller is assigned to each local and remote wavelength. They are referred to as local channel controllers (LCCs) and remote channel controllers (RCCs), as depicted in Fig. 1. Note that in the Figure we also show local and remote master con- trollers (LMCs and RMCs) which distribute clocking and perform other functions discussed below. The LCCs in a particular LON are interconnected via an electronic backplane bus which is used for exchange of low bandwidth status, configuration and synchronisa- tion information. The same is true for the RCCs. It can also be seen that via their special attachment to the local wavelength cross connect, the LMCs can commu- nicate among themselves using the remote reuse signal- ling function previously described.

Each user station contains two fixed tuned receivers plus a fast tunable transmitter, all capable of tuning across all channels. We assume that the number of stations would normally be greater than the number of local wavelengths, CL. Each station is pre-assigned a ‘Local HOME’ channel (LHOME) within the local waveband to which it fix tunes one of its receivers.

remote optical network

Fig. 1 Physical network architecture

248 IEE Proc -Commun., Vol. 144, No. 4, August 1997

I guard t ime+ preamble

Fig. 2 Loculfranzefovmut

Intra-LON operation can be viewed as an independent set of single-star networks, each using the AL wavelengths. Media access in this type of system has been widely investigated [I], and in this paper we consider a protocol which is similar to that of DCCN [5]. Protocol operation is divided into two levels which results in a dynamic movable-boundary TDMA scheme. At the higher level, each LCC allocates a contiguous block of data slots within the frame to each of the other LCCs in the LON, including itself. These blocks are used by stations on the LHOME channel of the LCC they were assigned to, for transmission to stations on the LHOME channel of the allocating LCC. Allocation information is exchanged among LCCs using the backplane bus. The lower level of media access is responsible for providing station access to individual data slots. Each frame on each channel contains a request subframe and an allocation subframe, as illustrated in Fig. 2. Within these, each station is assigned a request and allocation minislot on its LHOME channel. A request minislot consists of two fields. The DST field specifies the desired channel for transmission, and the NUM field specifies the number of slots requested. Similarly, an allocation minislot consists of three fields. The DST field specifies the channel a station may transmit onto, the START field specifies the starting data slot for transmission, and the NUM field specifies the number of contiguous data slots that may be transmitted. A station receives transmit access to one of the local wavelength channels by first tuning to its LHOME channel and then transmitting into its request minislot. The station then waits for an allocation in its allocation minislot on the LHOME channel, tunes its transmitter to the specified

IEE Proc-Commun., Vol. 144, No. 4, August 1997

channel and transmits the specified number of data slots. An LCC must continuously accept requests and generate slot allocations for all stations on its LHOME channel. A detailed hardware design capable of performing the request processing and allocation generation tasks in this type of system was described in [5 ] . The advantages of this design approach were discussed in [5] and comparisons were made with other options.

3 Remote media access protocols

In this Section we consider five remote media access protocols with varying degrees of coupling between local and remote system operation. A detailed descrip- tion of each is now given. Protocol I : Protocol 1 is the most obvious extension of the single-star case. The functionality for the remote media access protocol is duplicated from the local net- work. The same two level media access hierarchy and frame format are used. The functionality of the remote channel controllers is identical to that in the LONs. This protocol thus decouples system operation at the local and remote levels. However, owing to the large number of stations assigned to each RHOME channel the remote capacity can degrade because of request/ allocation overheads. In addition, RCC processing loads may become excessive. Protocol 2: Protocol 2 reduces the remote channel request/allocation signalling by completely eliminating it. This is achieved by pushing all remote signalling into the LONs. In this case, the remote frame contains only data slots. Each RCC allocates a contiguous block of slots to each LCC of each LON. Each station thus views a remote channel as simply another local channel and makes remote slot requests using its local request minislot. Similarly, a station receives remote slot allocations in its local allocation minislot. To simplify system operation we assume that all local and remote frames are aligned. It can be seen that the functionality of the remote channel controllers is greatly reduced for protocol 2, whereas the functionality of the local controllers increases. Protocols 3 and 4: In protocol 3, the same two level hierarchy and frame format as in protocol 1 are maintained. However, rather than assigning each station a dedicated remote request/allocation minislot pair, each LCC is assigned a set of remote request/ allocation minislots on the RHOME channel associated with its stations. A station must gain permission to use a remote request/allocation minislot pair before it may participate in the remote media access protocol. To do this, a station must execute a remote request arbitration protocol with its local channel controller. There are many possibilities for this protocol, here we use round-robin token passing with early token release. A station signals that it wishes to transmit using bits in its local request minislot. If the LCC is holding free remote request/allocation minislot pairs it will assign them to waiting stations. This notification is made in a station’s local allocation minislot. A station may use the remote request/allocation minislot pair for a predetermined maximum number of remote frames, or until it is no longer needed. The functionality of the remote channel controllers is identical to protocol 1, except that RCC processing loads are reduced due to this sharing, again at the expense of increased local

249

controller functionality. Protocol 4 is a minor variant of protocol 3 where minislot allocation requests are selected randomly. Protocol 5: Protocol 5 is a unique hybrid between pro- tocols 1 and 2. It increases remote capacity by pushing some of the signalling requirements from the remote channels onto the spatially reused local wavelengths over the RON. Each RCC allocates a contiguous block of slots within the frame to each other RCC, including itself. However, at the lower level each station is pre- assigned only a dedicated allocation minislot on its RHOME channel. Stations make remote channel slot requests via their dedicated local request minislot. The LCC passes the remote requests onto the associated HOME RCC via the reused local wavelength signalling network between the LMCs and the RMC. The func- tionality of the remote channel controllers is similar to that of protocol 1, however, they receive remote requests from the electronic backplane via the RMC rather than directly from the remote channels. Local channel controllers have the added functionality of sorting remote channel requests from local channel requests and passing them to the LMC via the local backplane.

4 Performance

In this Section we present both capacity and mean delay results for the five remote media access protocols. For all analyses, we assume the following default parameter values unless otherwise stated, CL = 15, CR

100, P = 5. If we assume that each channel is operating at a speed of lGbps and the remote request minislot length is 500 bits, then the station-star distance is 500m, the remote minislot duration is 0 . 5 ~ and the data slot duration is 2.5 ps.

= 5 , L = 5 , a = 0 . 5 , A = 5 , N s = 1000, D R ~ 2 0 0 , D q =

4. I Capacity performance In this Section we define several new capacity measures which are used in addition to conventional ones to compare the performance of each system. Remote channel capacity: This is defined as the fraction of time that the remote channels spend transporting full data slots. In each case, the computation is straightforward and the results are given as follows:

I-

System capacity: This is the sum of the local and remote protocol capacities, taking into account channel reuse. For protocol i it is given by

CAPS% CR * C A P R ~ + ACL . CAPL (5)

250

Maximum block capacity: This is a quantification of the ‘channel blocking’ associated with a specific protocol. It is defined as the maximum normalised throughput with which a station on one channel may communicate to a station on a destination channel. This occurs when all slot blocks except one on the destination remote channel are set to a size of one, while the final slot block is allocated the remaining data slots. For proto- cols 1, 3, 4 and 5, the maximum block capacity is easily seen to be CAPBMi = CAPR,(DR - CR + l)/DR, and for protocol 2 it is given by CAPBM2 = CAP,, (DL - ACL + l)/DL. Uniform block capacity: This is similar to the maximum block capacity except it occurs when all slot blocks on the destination remote channel are of equal size. This is a measure of the maximum throughput a station can use to communicate with a station on a destination channel under a uniform block allocation. For proto- cols 1, 3, 4 and 5, the uniform block capacity is easily seen to be CAPBUz = CAPRilCR, and for protocol 2 it is given by CAP,, = CAPR,/ACL.

0 2 1 6 8 10 12 11 16 18 20 remote channels

Fi .3

~ protocol 1 ~~~ protocol 2 _ _ _ _ protocols 3 and 4 . . . . . . . . . . . protocol 5

Remote channel capacity as a function of number of vemote chan- ne P s

O b i 1 6 8 IO 12 li 16 18 2b remote channels

Fi .4 n e t ~ protocol 1 ~~~ protocol 2 ~~~~ protocols 3 and 4

protocol 5

Maximum block capacity as a function ofnumber of remote chan-

. . . . . . . . . . .

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1 .o

0.8-

0 .6 - c .- U -

0 Q

I5t

\ \ ! \'::

\, \ \,':,,

oL 2 1 6 8 IO 1'2 1'L 1'6 1's 2b remote channels

System capacity as a function of number of remote channels Fig.6 ~ protocol 1 ~~~ protocol 2 _ _ _ _ protocols 3 and 4 .......... protocol 5

0 .6 - c .- U

a 0 . 1 -

Two sets of capacity results are presented, all with the network parameters set to the default values. The results are normalised to the capacity of a single remote channel. The first set, shown in Figs. 3-6, gives capacities as a function of the localiremote channel split. In this case, the total number of channels is fixed at 20, while the number of remote channels is varied. For protocols 3 and 4 the number of remote request minislots per remote channel is set to the minimum value of MR = [ACL/CR1. Fig. 3 shows the remote channel efficiency for protocols 1, 3, 4 and 5 increasing as CR increases owing to the decreasing number of sta- tions per RHOME channel. Protocol 2 has a decreas- ing remote channel efficiency for increasing CR since the remote frame structure is tied to the local one. Increasing CR decreases C,, therefore decreasing both local and remote channel efficiency. Fig. 4 shows the maximum block capacity of protocols 1, 3, 4 and 5 increasing rapidly with increasing CR and then remain- ing relatively constant. This is due primarily to rapidly increasing remote channel efficiency with increasing CR. All maximum block capacities are high owing to the small number of slot blocks, CR, per remote chan- nel. Protocol 2 has poor maximum block capacity owing to the high number of slots blocks, ACL, per remote channel. This effect also decreases with increas- ing CR owing to decreasing C,. Fig. 5 shows uniform block capacity for protocols 1, 3, 4 and 5 decreasing

rapidly as C, increases owing to an increased number of slot blocks per remote channel. Protocol 2 has very low uniform block capacity owing to large remote channel slot blocking. Fig. 6 shows the system capacity for all protocols decreasing as CR increases owing to reduced channel spatial reuse.

0.5 0 10 20 30 10 50

stations per Local channel

Fig.7 channel

Remote channel capacity as a function of number of stations per

~ protocol 1 _ _ _ ~ protocol 2 ~~~~ protocols 3 and 4 ............ protocol 5

1 0

0 8

h 0 6 e U 0

O L

0 2

t 1 01 " " " " ' I

0 10 20 30 LO 50 stations per local channel

Fig.8 Maximum block capacity as a function of number of stations per channel ~ protocol 1

protocol 2 ~ _ _ _ protocols 3 and 4 ............ protocol 5

~~~

0 .......,......... . .......,......... I ..... ...,......... .................. ........ < ......... 0 10 20 30 LO 50

stations per local channel

Fig.9 channel

~ protocol 1 ~~- protocol 2 ~~~- protocols 3 and 4 ........... protocol 5

Uniform block capacity us a function of nwnber of stations per

25 1 IEE Proc-Commun., Vol. 144, No. 4, August 1997

The second set of results, given in Figs. 7-10, shows capacities as a function of the number of stations per local channel. The number of remote request minislots per remote channel for protocols 3 and 4 is set to the minimum value. Fig. 7 shows remote channel efficiency for protocols 1, 2 and 5 decreasing as N increases owing to the increased overhead per remote and local channels. Protocols 3 and 4 are insensitive to increasing N since the existing remote request/allocation overhead on the remote channels can be shared at the cost of increased access delay. Fig. 8 shows that the maximum block capacity of protocols 1 and 5 decreases with increasing N owing to decreases in remote channel effi- ciency. Protocols 2, 3 and 4 remain essentially insensi- tive to increasing N. Fig. 9 shows that the uniform block capacity for protocols 1, 3, 4 and 5 is very low but affected only by remote channel efficiency as N increases. Protocol 5's uniform block capacity remains low owing to a large number of slot blocks, but is insensitive to increasing N. Fig. 10 shows the system capacity for all protocols decreasing only slightly as N increases owing to decreased local and remote channel efficiencies. Spatial channel reuse remains constant as N increases.

i

sot i :: LO " 1

0 10 20 30 LO 50 0

stations per Local channel

Fig. 10 __ protocol 1 _ _ _ protocol 2 _ _ _ _ protocols 3 and 4 . . . . . . . . . . . . protocol 5

System capacity as afunction oftlumber of stutionsper chunnel

In the capacity results presented, it is apparent that the remote capacity may be much larger in protocols which couple local and remote system operations. This is attained without any noticeable reduction in local capacity. The improvement in remote capacity is significant since these channels are not reused spatially, and thus it is desirable to use each as efficiently as possible. In addition, maximum and uniform block capacity may also be improved by coupling the protocols.

4.2 Mean delay Discrete event simulations were used to analyse and compare the mean access delay associated with each remote media access protocol. This delay is defined to be the time from the arrival of a packet until it commences transmission. The remote channel slot block allocations were assumed to be fixed, and results were collected for traffic destined to a single remote channel. It was also assumed that no bit errors occur during transmission, and packet arrival rates are Poisson distributed with packet lengths equal to one data slot. Two sets of experiments were performed,

both with the network parameters set to the default values previously described. All experimental data points were obtained through simulation experiments of lo5 remote frames and repeated with different random generator seeds. For the first set of experiments, 10 stations generated uniform traffic load to a single destination remote channel. A slot block of 100 remote data slots was used for protocols 1,3,4 and 5 , while protocol 2 used a slot block of 25 data slots. Plots were generated showing the remote access delay, normalised to the remote request minislot duration, versus total throughput, normalised to the capacity of the destination channel. Figs. 11-15 illustrate the uniform access delay for Protocols 1 through 5 . Results for Protocols 1, 2 and 5 show access delay versus throughput curves with similar shapes. These curves show a low, relatively constant access delay which extends throughout a range of throughput, then a rapid increase in delay. Protocols 3 and 4 exhibit slightly different behaviour due to sharing of the remote request/allocation minislots. This is a function of the remote request to active station ratio and the remote request hold time. The plots of Figs. 13 and 14 show the cases of R = 8 and R = 5 with N = 10 and H = 2. With R approaching N the delay curves tend to move towards that of protocol 1, while as R decreases with respect to N, the access delay increases approximately linearly with increasing throughput up to a saturation point, at which time it increases rapidly. We can also observe a decrease in achieved throughput owing to the additional overhead as more remote request/allocation resources are added.

throughput, %

Fig. 11 Uniform loading access delay, protocol 1 (slot block size = 100)

30001

throughput, %

Fig. 12 Uniform loading access delay, protocol 2 (slot block size = 25)

252 IEE Proc.-Commun., Vol. 144, No. 4, August 1997

5000

L 000

3000 x 0 W d

2000

1000

0 0

throughput,%

Fig. 13 -e-

UnEform loading access delay, protocol 3 (slot block size = 100) R = 5 , H = 2

--+-- R = 8 , H = 2

I

throughput, Yo

t 1

Fig. 14 -*- Uniform loading access delay, protocol 4 (slot block size = 100) R = 5, H = 2 R = 8 H = 2

> , > , 1 1 1 . -

throughput, % Fig. 15 Uniform loading access delay, protocol 5 (slot block size = 100)

For the second set of experiments, 10 stations generated skewed traffic to a single destination channel. Among the stations, 8 were very lightly loaded with approximately 5% of the slot block capacity and remained so for the experiment, while 2 of the stations had their load increased from very low load to very high load. Plots were generated showing the remote access delay versus throughput for the low fixed-load stations and both of the stations with variable load. In this case, the throughput was that used by the individual station normalised to the slot block capacity allocated on the remote channel. In the interests of

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1 5 10 15 20 25

arr ival rote, slots/fmme Fi .16 Simulation-theory comparison, protocol I (tRF = 1300) 2- simulation - -0- - Q-theory

8000 ,

arr ival m te, slots / f ro me

Fig. 17 -e- simulation - -0- - Q-theory

Simulation-theory comparison, protocol 2 ( t L F = t R F = 521)

The results of the analytic/simulation comparison are illustrated in Figs. 16-20. The modelling results for protocols 1, 2 and 5 compare very closely with the

253

10000 -

,,OOt 8/

8000

x

aJ U Z 6000- I -

Fig. 18 -4- simulation

Simulation-theory comparison, protocol 3 (tRF = 1090)

2000

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-

12000

10000

8000

2000-

t 1

L J

I 2000 -

I O b 5 10 15 20 25

arrival rote,slots/frame

Simulation-theory comparison, protocol 5 (tRF = 1300, tLF = Fig.20 521) -4- simulation - -0- - Q-theory

5 Conclusions

In this paper, we have considered a passive WDM opti- cal network which incorporates hierarchical wavelength spatial reuse. Such an architecture can increase system capacity by exploiting the natural traffic locality found in real LANs and MANS. The architecture is based on a local/remote hierarchy and requires that each user station contain two receivers and a single agile trans- mitter. This maintains many of the desirable features of a passive network and single-hop communication.

Five remote media access protocols were introduced, analysed and compared. All protocols were based upon the use of remote reservations with variations on how stations gain access to the reservation resources. Three of the protocols make use of dedicated remote reserva- tion resources while two were based upon sharing. Pro- tocol 1 is a straightforward extension of conventional single star protocols and leads to a decoupling of remote and local system operation. Protocols 2-5, how- ever, use various forms of coupling between local and remote system operation. It has been demonstrated that, by coupling the media access at both levels, the remote channel efficiency is improved with varying effects on the average remote access delay and delay coupling experienced by a station.

6 Acknowledgment

This work was supported by the Telecommunications Research Institute of Ontario (TRIO).

7 References

1 MUKHERJEE, B.: ‘WDM-based local lightwave networks, part 1: single-hop systems’, IEEE Netw., 1992, 6, (3), pp. 12-27

2 JANOSKA, M., and TODD, T.D.: ‘A single-hop wavelength- routed local area network‘. IEEE international conference on Applications of photonics technology, Toronto, 1994

3 DOWD, P.W., BOGINENI, K.K., ALY, K.A., and PER- REAULT, J.A.: ‘Hierarchical scalable photonic architectures for high-performance processor interconnection’, IEEE Trans. Com- put., 1993,

4 “ N A N . B.. FOTEDAR. S.. and GERLA. M.: ‘A two level optical st& WDM metrol;&fan area netwbrv. Proc. IEEE GLOBECOM, 1994, pp. 563-566 JANOSKA, M.W., and TODD, T.D.: ‘A simplified optical star network using distributed channel controllers’, J. Lightwave Tech-

5

nol., 1994, 12; (l l) , pp. 2011-2022 6 HUANG, N.-F., WU, C.-S., and MA, G.-K.: ‘A time-wavelength

scheduling algorithm for interconnected WDM star networks’. Proc. IEEE INFOCOM, 1994, pp. 552-559 MATSUMOTO, T.: ‘Multiple-access optical network architecture employing a wavelength and network division technique: MAN- DALA’. Proc. IEEE GLOBECOM, 1993, pp. 463467

8 HAYES, J.F.: ‘Modeling and analysis of computer communica- tions networks’ (Plenum Press, New York, 1984)

9 TODD, T.D., and BIGNELL, A.M.: ‘Traffic processing algo- rithms for the SIGnet metropolitan area network’, IEEE Trans. Commun., 1992, 40, (3), pp. 568-576

I

7

8 Appendix: Analytic delay model

Additional details for the queueing model are presented here. The development is divided into two sections, the calculation of f i which is the average wait of a tagged packet within the queue, and the determination of the protocol specific fixed delay. The determination of m is motivated by the model first proposed in [5]. System time is divided into remote frames with a number of data slots per remote frame. Arrivals to the global queue are Poisson with a rate of A packets per frame, while departures are deterministic at a rate of b packets per frame. Departures are assumed to occur at the beginning of each frame. The objective is to calculate

IEE Proc.-Commun., Vol. 144, No. 4. August 1997 254

mathematically the average queueing delay experienced by a packet. We define ni as the number of enqueued packets at the end of frame i, b as the number of packets removed per frame, Pj as the probability of j packets enqueued at the end of a frame and a, as the number of packets arriving during frame i. The analysis begins by embedding a Markov chain at the end of each frame. The state equation for the number of pack- ets in the queue can easily be written as ~ l , + ~ = max(n, - 6 , 0) + a,+l. Using this, the probability generating func- tion N(z) of n, is found to be

Note that Rouche’s theorem [8] can be used to solve for Pi, i E (0, ..., b - 1) by forming an appropriate set of linear equations. Other values of Pi may be found by taking the inverse Fourier transform of N(z). This can be calculated numerically, as described in [8] by evalu- ating N(z) at a sufficiently large number of points around the unit circle in the complex z-plane.

Using a similar analysis as in [SI, we can derive the delay experienced by a ‘tagged’ packet arrival. Define no as the number of packets enqueued at the start of an arbitrary frame. At some time T after the start of the frame our tagged packet arrives. At the start of the next frame the number of packets enqueued in front of the tagged packet, including the tagged packet, is equal to the number carried over from the previous frame, nol, and the number of new packets, a , arriving in the interval z, i.e. n1 = nol + a, + 1. If we define m as the number of frames required to service n1 packets then E(m) is the average number of frames. To determine this we need to know P,(i), the PDF of m. We first consider P,(i) conditioned on nl . After removing the conditioning and after some straightforward manipulation, the value of I% can be calculated. The total delay consists of the queueing delay plus the fixed delays due to station-star propagation delay and, for protocols 3 and 4, sharing of remote request minislots. In the following development, a number of simplifying approximations are made. Protocols 1 and 2: Fixed delays are due to station-RCC request signalling, alignment delay at the remote

channel controller and RCC-station propagation delay. This can be expressed as & = tRF + 2P - P mod tRF + tRFI%. Prom an access delay viewpoint, protocol 2 is identical. Protocol 3: In this case, the inter-remote request transmission points do not occur at periodic intervals of tRF but rather at intervals of t&di. Here % is the average number of remote frames between a station being able to make use of a remote request minislot. The average delay can be expressed as & = (tRd2)(E + 1) + 2P ~ P mod tRF + t & % . A reasonable estimate of f i can be calculated as E = (nmin + nm&2 = 1 + H . max(N ~ R, 0)/2. Actually, n is a function of the remote slot block size b and the aggregate arrival rate h due to early release of the remote request minislot. Since protocol 3 uses the stations local request/ allocation minislot to execute the remote request minislot sharing protocol, this is only reasonable when

Protocol 4: The expression for is identical to that of protocol 3. The difference is in the calculation of n since remote request/allocation minislots are assigned randomly. The analysis assumes that the probability that a station waits k frames for an assignment is geo- metrically distributed with parameter Pa, equal to the probability of assignment in a particular frame. We cal- culate this by first considering Pa;, which is the condi- tional probability that a request is assigned to a station in the frame, given that i requests are available for assignment. This is uniformly distributed but depend- ent upon N and R. In order to remove the condition from Pai we require the determination of Pri, the prob- ability that the local channel controller has i remote requests to assign. If we let P,, the probability of a par- ticular remote request being available, be liH, then P,, is binomially distributed. Following some straightfor- ward simplification, n can easily be calculated. As in protocol 3, this result is reasonable when t,&tRF < 112. Protocol 5: The model for protocol 5 is slightly differ- ent owing to the use of local requests and remote allo- cations. However, using a similar argument to that above, the average access delay may be expressed as 6

tLJdtRF < 112.

= 2P + t&2 f tRFm when tLF < tRp

IEE Proc -Commun., Vol. 144, No. 4, August 1997 255