llc-mac analysis of general packet radio service in gsm

Bell Labs Technical Journal ◆ July–September 1999 37

IntroductionThe need to support voice services and user

mobility has led to standardization of one of the most

popular digital cellular standards, the Global System

for Mobile Communications (GSM), by the European

Telecommunications Standards Institute (ETSI). In

addition to speech applications, GSM can provide

some circuit-switched data services such as facsimile

transmission and short messaging services (SMS).

However, SMS volume is expected to exceed the GSM

capacity in two to five years. Due to recent develop-

ments in mobile computing devices and the emerging

market of multimedia communications, there is an

increasing demand for more advanced data service

providers in GSM. This has led to standardization of

two new data-bearer services as part of GSM

Phase 2+, namely, High-Speed Circuit-Switched Data

Service (HSCSD) and General Packet Radio Service

(GPRS), which operate in circuit-switched and packet-

switched modes, respectively. HSCSD allows a mobile

station (MS) to access more than one radio channel

operated in circuit-switched mode simultaneously,

thereby increasing the MS transmission rate to

64 kb/s. For applications generating bursty traffic, a

packet-switched bearer is more bandwidth efficient

than a circuit-switched one. Thus, the purpose of GPRS

is to statistically multiplex bursty data traffic efficiently.

GPRS technology is overlaid on the existing GSM

infrastructure. GPRS and GSM services share dynami-

cally the same radio resources and the same radio

channel structure. The GPRS network is seen as the

packet-switched extension of GSM. In GPRS, up to

eight users can be dynamically multiplexed on the

same radio channel if they require low bit rates, and

eight radio channels can be allocated to one user with

high-bandwidth requirements.1,2 Thus, GPRS provides

flexible bandwidth allocation and is economical to

users. The demand for voice is almost saturated, while

multimedia traffic is going to explode; therefore, the

evolution of the GPRS platform is seen as a lead-in to

third-generation wireless communication.

In order to maximize user data throughput and

decrease data-packet transfer time, the GPRS protocol

stack3,4 shown in Figure 1 should be optimally imple-

mented. In this figure, we highlight in color the proto-

♦ LLC-MAC Analysis of General Packet RadioService in GSMCristian Demetrescu

General Packet Radio Service (GPRS) is one of the major services currently standard-ized by the European Telecommunications Standards Institute (ETSI) for GlobalSystem for Mobile Communications (GSM) Phase 2+. GPRS has been introduced tosupport bursty packet-switched traffic such as e-mail, World Wide Web traffic, andtelemetry. This paper presents a unified analytical model of two GPRS protocols,namely, logical link control (LLC) and radio link control/medium access control(RLC/MAC). The interaction between selective automatic repeat-request (ARQ)schemes at the two layers is investigated. In order to save scarce radio resources, wecalculate a finite optimal number of RLC/MAC retransmissions per LLC frame as afunction of the number of radio link errors and the LLC frame size. A novel channel-dependent ARQ mechanism is introduced at the LLC layer. Finally, throughput anddelay performances of the two ARQ stacks are presented.

38 Bell Labs Technical Journal ◆ July–September 1999

col layers under consideration in this paper—that

is, the logical link control (LLC) and the radio link

control/medium access control (RLC/MAC) layers.

In order to protect LLC peer-to-peer communica-

tion, packet data units (PDUs) coming from higher lay-

ers are segmented into variable-size LLC frames, and 24

cyclic redundancy-check (CRC) bits are added to every

LLC frame for error detection. A stop-and-wait auto-

matic repeat-request (ARQ) mechanism is implemented

at the LLC to retransmit any erroneous LLC frames,

rendering to the LLC peer-to-peer link a high

reliability.3 Each LLC frame is passed to the RLC/MAC

layer, where it is further segmented into RLC/MAC

blocks of fixed size depending on the channel coding

scheme (CS) used at the physical layer.1,5 The

RLC/MAC layer provides a selective ARQ mechanism

for further error recovery over the radio interface (Um).

This paper describes the interaction between the

two ARQ mechanisms at the LLC and RLC/MAC lay-

ers, respectively. We assume that the LLC layer works

in the acknowledgment (ACK) mode so each LLC

frame is acknowledged upon receiving it. The main

objective of this work is to define analytically closed-

form expressions for the packet transfer delay and the

throughput that can be used for on-line quality-of-

service (QoS) management. The packet transfer delay

and the throughput seen at the LLC layer are two of

the most important parameters of the QoS profile;6

their on-line estimation is crucial for network perfor-

mance. We present a novel derivation of the ARQ

parameters (that is, the number of retransmissions) at

the LLC and RLC/MAC layers that strike a balance

between the bandwidth efficiency within the GPRS

access network and the efficiency over the Um.

Most of the papers published on LLC-RLC/MAC

performance analysis in GPRS7-12 make use of numer-

ical simulations to estimate the LLC throughput and

the LLC transfer delay when the LLC works in non-

acknowledgment mode. Ludwig and Turina12 derived

a formula for the average LLC frame transfer delay at

the RLC/MAC layer using an iterative method to cal-

culate the RLC/MAC retransmission probability. They

assumed Poisson arrival traffic and an infinite number

of RLC/MAC retransmissions per LLC frame.

However, their assumptions are general and do not

account for the LLC protocol.

In this paper, we derive closed-form expressions

for the LLC frame transfer delay and the throughput at

the LLC layer (irrespective of the arrival traffic) under

more realistic assumptions—namely, a finite number

of retransmissions per LLC frame and a two-state radio

channel model. This analytical model of the stack of

the two ARQ mechanisms at the LLC and RLC/MAC

layers is, to the best of the author’s knowledge, pre-

sented for the first time in this paper. Further, in order

to optimize radio resources, a channel-dependent ARQ

mechanism is introduced at the LLC layer.

Panel 1. Abbreviations, Acronyms, and Terms

A-bis—BTS/BSC interfaceACK—acknowledgmentARQ—automatic repeat requestBLER—block error rateBSC—base station controllerBSS—base station subsystemBTS—base transceiver stationCDF—cumulative distribution functionCRC—cyclic redundancy checkCS—coding schemeETSI—European Telecommunications Standards

InstituteGb—BSS/SGSN interfaceGGSN—gateway GPRS support nodeGPRS—General Packet Radio ServiceGSM—Global System for Mobile CommunicationsHSCSD—High Speed Circuit-Switched DataIP—Internet protocolLLC—logical link controlMAC—medium access controlMS—mobile stationNACK—negative acknowledgmentPDF—probability density functionPDU—packet data unitQoS—quality of serviceRLC—radio link controlSGSN—serving GPRS support nodeSMS—short messaging servicesSNDCP—subnetwork dependent convergence

protocolSN-PDU—SNDCP-PDUTCP—transmission control protocolUm—radio interface


The section “ARQ Mechanism at RLC/MAC”

describes an analytical approach used to derive the

probabilities of RLC/MAC retransmissions when the

radio channel is modeled as a two-state Markov

process. This analysis gives us a closed-form expression

of the most likely number of RLC/MAC retransmis-

sions as a function of the RLC/MAC block error rate

(BLER) and the LLC frame size. The most likely num-

ber of RLC/MAC retransmissions can be used to esti-

mate the LLC frame transfer delay and the LLC

throughput as well as the signaling overhead required

by the ARQ mechanisms at both the RLC/MAC and

the LLC layers. This RLC/MAC protocol improves the

radio bandwidth efficiency by limiting the number of

RLC/MAC retransmissions per LLC frame when the

radio channel is in a “bad” state.

In the section “Channel-Dependent LLC

Protocol,” we introduce a novel LLC protocol that

accounts for the radio channel status. By means of a

time-out error-recovery algorithm, the ARQ mecha-

nism at the LLC layer attempts retransmission of an

erroneous LLC frame only after a time interval, intro-

ducing a time diversity.13 This channel-dependent

ARQ mechanism14 also increases MS battery lifetime

by deferring transmission over a “bad” channel to a

later time when the channel is likely to be in a “good”

state. The section “LLC Throughput” presents analyti-

cal formulas for both the throughput and the delay per

Application

IP/X.25

SNDCP

LLC

RLC

MAC

GSM RF

MS

Relay

RLC

MAC

GSM RF

BSSGP

Frame relay

L1bis

BSSUm Gb

LLC

SGSN

BSSGP

Frame relay

IP

L2

L1bis L1

SNDCP GTP

IP

L2

L1

GTP

Gn

IP

GiGGSN

BSS – Base station subsystemBSSGP – BSS GPRS protocolGb – BSS/SGSN interfaceGGSN – Gateway GPRS support nodeGi – Interface between two GGSNsGn – SGSN/GGSN interfaceGPRS – General Packet Radio ServiceGSM – Global System for Mobile CommunicationsGTP – GPRS tunnel protocolIP – Internet protocolITU-T – International Telecommunication Union— Telecommunicaton Standardization SectorL1 – Layer 1 for Gn interface

L1bis – Layer 1 for Gb interfaceL2 – Layer 2 for Gn interfaceLLC – Logical link controlMAC – Medium access controlMS – Mobile stationRF – Radio frequencyRLC – Radio link controlSGSN – Serving GPRS support nodeSNDCP – Subnetwork dependent convergence protocolUm – Radio interfaceX.25 – ITU-T standard for network layer 3

Figure 1.GPRS transmission plane.


LLC frame. These closed-form expressions, together

with measurements, are suitable for on-line GPRS

implementation. Several samples of numerical simula-

tions are presented to illustrate the analytical formulas.

Throughout this work, denotes the BLER,

denotes the LLC frame size in RLC/MAC blocks,

is the maximum number of RLC/MAC block retrans-

missions per LLC frame, and is the maximum num-

ber of LLC retransmissions per LLC frame. Henceforth,

the expression “RLC/MAC retransmission per LLC

frame” denotes the retransmission of several

RLC/MAC blocks belonging to the same LLC frame

that have been reported as being in error by a negative

acknowledgment (NACK). These blocks are selectively

repeated. An LLC retransmission occurs only when

the ARQ mechanism at the RLC/MAC layer fails to

correct all the errors of an LLC frame within the maxi-

mum allowed number of RLC/MAC retransmissions.

ARQ Mechanism at RLC/MACWe assume that a radio channel is described by

the Gilbert-Elliott model, which is a two-state Markov

process.15 The two channel states are denoted by

“bad” (where the BLER is equal to ) and “good”

(where the BLER is zero), respectively. Rigorously

speaking, the BLER in the good state is not zero but is

very small compared with that from the bad state.

Since the dominant degradation of network perfor-

mance comes from errors in the bad state, the effect of

errors in the good state on network performance can

be neglected.

Throughout this analysis, we consider that the

errors over the radio channel in a bad state are inde-

pendent. This is an optimistic assumption because, in

general, the errors over the radio channel occur in

bursts—exhibiting, therefore, some degree of correla-

tion. In a bad state, the errors may affect several

RLC/MAC blocks in sequence. However, the probabil-

PB

L

MN

PB

Panel 2. Mathematical Variables and Expressions

δ (k)—Dirac function—standard deviation of k—standard deviation of —carrier to interference ratio

E [k]—average number of RLC/MAC retransmissionsper LLC frame

E [R]—average number of retransmissions perRLC/MAC block

HLLC—number of header octets in an LLCframe

HRLC/MAC—number of header octets in anRLC/MAC block

k—number of RLC/MAC retransmissions per LLCframe

L—maximum number of LLC retransmissions perLLC frame

LLC timer—timer that is set when an LLC frame issent to the RLC/MAC layer

m—number of NACKs sent per LLC frameM—maximum number of RLC/MAC block

retransmissions per LLC frameM(N)—number of NACKs sent per LLC frame of

size equal to N RLC/MAC blocksN—number of RLC/MAC blocks in an LLC framenLLC—number of information octets in an LLC

frame

nRLC/MAC—number of data-information octets inan RLC/MAC block

PB—non-zero BLERPM—probability that after RLC/MAC retrans-

missions there are still RLC/MAC blocks in errorbelonging to the same LLC frame of size

Pr—probabilityPrc—conditional probabilityPr( ≤ R)—probability that the radio channel is

in a bad stateq—probability of successful transmission of an

LLC frameQ—number of LLC frames queuing for transmis-

sion at the RLC/MAC layerR— threshold, 12 dBThroughputLLC—throughput at the LLC layerTLLC Frame Service Time—average service time of an

LLC frameTLLC RLC/MAC—service time of an LLC frame at the

RLC/MAC layerTLLC Service Time—service time of an LLC frame at

the LLC layerTNACK—time required to receive and process a

negative acknowledgment, 40 msTRLC/MAC Block—transmission duration of an

RLC/MAC block over the radio interface, 18.46 ms

C/I

C/I

N

MC/IC/Is0

s


ity of a deep Rayleigh fade lasting more than 20 ms

(that is, more than one RLC/MAC block) is very low,

so we neglect the correlation of errors over many

RLC/MAC blocks.

We assume that the BLER is constant over the

transmission of an LLC frame. This is a reasonable

assumption, since the BLER is derived from a lognormal,

slow-fading distribution of the carrier-to-interference

ratio, C/I, for which a slow fade can last several sec-

onds. We proceed now to calculate the following con-

ditional probabilities:

, (1)

conditioned by the radio channel being in a bad state

(that is, by ). The notation refers to

RLC/MAC blocks belonging to the same LLC frame.

Note that, when the channel is in the good state, no

retransmission is required as we have assumed that

the BLER is zero in this state. The probability that the

channel is in a bad state is given in (7).

The probability that all RLC/MAC blocks

of an LLC frame are received error free at the first

transmission is equal to

, (2)

where we assume that the transmissions of all

RLC/MAC blocks are statistically independent events.

To calculate , we note that the probability that

an RLC/MAC block has errors after two retransmis-

sions is ; therefore, the probability that this

RLC/MAC block is successfully received after two

RLC/MAC retransmissions (including the original

transmission) is . Using this and the independence

of the transmissions of the RLC/MAC blocks yields

(3)

where and are the probabilities that

after the second and first RLC/MAC transmissions,

respectively, the LLC frame is received error free.

Thus, is the probability that two RLC/MAC

transmissions are required to send the LLC frame

because after the first RLC/MAC transmission there

are still erroneous RLC/MAC blocks. Following the

above approach, the general form of is

. (4)

From (4) we have

(5)

and

. (6)

It is worth pointing out that the conditional probabil-

ity, , from (4) is a closed-form expression.

Ludwig and Turina12 have also derived , but in

an implicit, iterative form. The equivalence between

the closed-form expression of given in (4) and

that given by Ludwig and Turina is shown in the

appendix. The benefit of deriving a closed-form

expression for is that we can analytically derive

the most likely number of RLC/MAC retransmissions

per LLC frame, , for a given and a given LLC

frame size, as shown shortly.

To remove the condition on the channel state

from , we use the probability that the radio

channel is in a bad state

(7)

to obtain the unconditional probability of the

form

(8a)

and

(8b)

where is the threshold that delimits good and bad

states (12 dB), , and . Here, x

represents and �0 denotes the standard deviation of

. In (8a), the term represents the prob-

ability that an LLC frame is transmitted over a channel

in a good state and that, therefore, no retransmission is

required. Using the tabulated integral16 from (7), we

1-Pr1C@I# R2C@I

C@I

s0 5 7 dBC@I5 16 dB

C@IR

Pr Pr forB Bk R P P kCI

kN

kN

( ) = ≤( ) −( ) − −( )

≥−1 1 21

Pr Pr Pr forB1 1 1 1( ) = ≤( ) −( ) + − ≤( ) =CI

NC

IR P R k

Pr1k2

Pr1C@I# R251

s0" 2p3

R

-̀

expc-1x-C/I22

2s20

ddx

PrC1k2

PBkopt

PrC1k2

PrC1k2

PrC1k2

PrC1k2

a

`

k5 1

311-PkB2N-11-Pk-1B 2N45 1

limPr limC B Bk k

kN

kN

k P P→∞ →∞

−= −( ) − −( )

=( ) 1 1 01

PrC1k25 11-PkB2N-11-Pk-1B 2N

PrC1k2

PrC122

11-PB2N11-P2B2

N

PrC1225 11-P2B2N-11-PB2

N

N

1-P2B

P2B

PrC122

N

PrC1125 11-PB2N

NPrC112

PrC1k2BLER 5 PB

retransmission ; PrC1k2

in error that are corrected by the kth RLC/MAC

retransmission there are RLC/MAC blocks

Probability that after the 1k-12th RLC/MAC


obtain . From (8), we note that

.

The number of RLC/MAC retransmissions per LLC

frame, , is a random variable with a probability den-

sity function (PDF) equal to where is

the Dirac function and is given in (8). Then, the

average value of is

(9)

and its variance has the form

(10)

where is the maximum number of RLC/MAC

retransmissions per LLC frame defined in (12) and is

the standard deviation of .

Using (4) through (6), we can also define a con-

ditional PDF for , namely, . As we are

interested in the number of RLC/MAC retransmis-

sions when the radio channel is bad, we plot in

Figure 2 this conditional PDF and the corresponding

conditional cumulative distribution function (CDF)

for various LLC frame sizes expressed in RLC/MAC

blocks. These data are for the case where

and .

The average and standard deviation of random

variable can be obtained using (9) and (10), respec-

tively. For , , and , we obtain

and . We can see from Figure 2a that

there is a maximum value of the , which will be

analytically derived next.

The most likely number of RLC/MAC retransmis-

sions per LLC frame—that is, the value of that maxi-

mizes —can readily be derived by differentiating

with respect to in (4) or (8) and setting the obtained

result equal to zero, yielding

(11)

where the square brackets [ ] denote the closest inte-

ger less than or equal to the term inside the brackets.

It can be seen from (11) that the ratio of the two

logarithm functions inside the square brackets is posi-

k

P

P

P

N

N N

opt

B

B

B

ln

ln=

−

−

+

−( )−( )

1

11

1 1

1

k

PrC1k2

k

PrC1k2

s 5 0.9k 5 2.5

M 5 7PB 5 0.2N 5 20

k

M 5 7

PB 5 0.2

PrC1k2d1k2k

k

s

M

E31k-k2245 s2 5 a

M

k5 1

k2Pr1k2

E3k45 k 5 a

M

k5 1

kPr1k2

k

Pr1k2

d1k2Pr1k2d1k2

k

a

`

k5 1

Pr1k25 1

Pr1C@I# 1225 0.3

0.00

0.10

0.20

0.30

0.40

0.50

0.60

1 2 3 4 5 6 7

RLC/MAC retransmissions

0.00

0.20

0.40

0.60

0.80

1.00

1.20

1 2 3 4 5 6 7

RLC/MAC retransmissions

Co

nd

itio

nal

PD

F

Co

nd

itio

nal

CD

F

N = 20N = 50N = 100

(a) (b)

CDF – Cumulative distribution functionLLC – Logical link controlMAC – Medium access controlN – LLC frame size in RLC/MAC blocksPDF – Probability density functionRLC – Radio link control

Figure 2.Statistics of the RLC/MAC retransmissions for and .M 5 7PB 5 0.2


tive and therefore . From (11), we can derive

two asymptotic results for , one when (that

is, for a large LLC frame) and the other when

(that is, for a small LLC frame). For large LLC frames

we have the limit while for small LLC frame

size the limit becomes , as expected. As an

example, we insert and into (11) to

obtain , which is in agreement with the

numerical results shown in Figure 2a. Thus, the most

likely number of RLC/MAC retransmissions per LLC

frame given in (11) is a function of the size of the LLC

frame, , and the block error rate, . The latter

dependency is further illustrated in Figure 3 for differ-

ent values of . Figure 3 shows that for small the

most likely number of RLC/MAC retransmissions is

not very sensitive to the LLC frame size, while for

larger this sensitivity increases.

Other limiting cases can be derived from (11)

when and , respectively. When goes to

zero in (11) we obtain ; for , we have

. Thus, for either a very bad radio environ-

ment (that is, ) or for a large LLC frame size

(that is, ), becomes very large. In practice,

when is larger than a predefined error-rate thresh-

old,1,5 either a handover is initiated or, if this is not

possible, a radio-status PDU is sent to the LLC layer to

inform it about the bad status of the radio interface.

When the radio channel is bad (that is, when is

large), a good practice is to defer the RLC/MAC trans-

mission to a later time when the radio channel is good

rather than retransmitting erroneous RLC/MAC

blocks, therefore wasting MS battery lifetime.

It can be seen from Figure 2b that the CDF of

converges very quickly to unity. For ,

, and , the CDF is already

equal to 0.8. In this case, 80% of the RLC/MAC blocks

of an LLC frame are received error-free after only

three retransmissions. Therefore, one can use the most

likely value, , of the number of RLC/MAC retrans-

missions per LLC frame as a measure for the maxi-

mum allowed number of RLC/MAC retransmissions,

, per LLC frame. In a GPRS implementation, could

be chosen as

. (12)

The calculation of can be done on line based on

(11), the measured value of , and the LLC frame size

in RLC/MAC block units. It is worth pointing out that

PB

kopt

M 5 kopt

MM

kopt

k 5 kopt 1 1 5 3N 5 20

PB 5 0.2PrC1k2

PB

PB

kN W 1

PB S 1

kopt S `

PB S 1kopt 5 1

PBPB S 0PB S 1

PB

PBPB

PBN

kopt 5 2

N 5 20PB 5 0.2

kopt S 1

kopt S `

N S 1

N W 1kopt

kopt $ 1

012

3456

78

2 22 42 62 82

LLC frame size (N)

k op

t

PB = 0.2PB = 0.5

102

kopt – Most likely number of RLC/MAC retransmissions per LLC frameLLC – Logical link controlMAC – Medium access controlPB – Block error rateRLC – Radio link control

Figure 3.The most likely number of RLC/MAC retransmissions per LLC frame.


the maximum allowed number of RLC/MAC retrans-

missions per LLC frame given in (12) balances the user

throughput and the use of the radio channel.

Alternatively, one can use the CDF of to derive

as follows. The CDF of the random variable can be

derived from (8) in the form

. (13)

As a system requirement, we assume now that the

probability of successful transmission of an LLC frame

is equal to . With this system requirement, (13)

becomes

(14)

yielding

(15)

where . Both (12) and (15) can

be used to calculate the maximum allowed number of

RLC/MAC retransmissions per LLC frame.

Channel-Dependent LLC ProtocolAn accurate estimation of the optimum number of

RLC/MAC retransmissions per LLC frame ( ) is of

crucial importance in managing network resources.

For example, once has been estimated using either

(12) or (15), the transmission delay and the through-

put per LLC frame can be evaluated as shown below.

This information can then be used by the admission

control unit to define the QoS profile provided by the

network to a customer. At this point, the value of the

LLC timer that guards this LLC peer-to-peer connec-

tion is set.

To ease the readability of the paper, we now

define several notations used henceforth. We denote

by the service time of an LLC frame at the

RLC/MAC layer and by the service time of

an LLC frame at the LLC layer. In addition, we denote

by the total number of LLC frames queuing for trans-

mission at the RLC/MAC layer. Then the LLC timer

(that is, the timer set when an LLC frame is sent to

the RLC/MAC layer) can readily be derived using

from (12) or (15) as follows:

(16)

where

(17)

and

. (18)

Here, denotes the average number of retransmis-

sions per RLC/MAC block; is the trans-

mission duration of an RLC/MAC block over the radio

interface (18.46 ms); is given in (9); and is

given in (12). , which is the time to receive and

process a NACK (40 ms), includes the NACK/ACK

processing delay at both the MS (10 ms) and the base

station subsystem (BSS) (10 ms) as well as the

NACK/ACK transfer delay (20 ms) between the MS

and the BSS. In (16), the term is a positive constant

introduced in order to avoid the fade on the radio

channel at the next LLC retransmission. Thus, for

a time diversity is provided by the LLC time-out

recovery mechanism. In practice, is set by measure-

ments taken on the radio channel. For brevity, we

have included the radio-access delay in the term in (16).

It is worth pointing out that the LLC frame queu-

ing delay at the RLC/MAC layer—that is, the term

in (16)—provides a natural, intrinsic time diversity.

Thus, if an LLC transmission fails, the LLC will retrans-

mit the erroneous LLC frame to the RLC/MAC layer.

This LLC frame should wait in the RLC/MAC queue

until all LLC frames in front of it are served. This

queuing delay introduces an internal time diversity.

However, under a light and bursty traffic load, the

queuing delay at the RLC/MAC layer could be very

small so that there is a need for the additional time

diversity introduced by the term in (16).

The LLC frame queuing delay at the RLC/MAC

queue is explicitly taken into account in (16) by the

presence of the factor . The estimation of is based

on the measured number of LLC frames queuing at

the RLC/MAC queue and therefore is accurate on

average, irrespective of the LLC frame arrival process.

The expression (16) can be computed on line using

QQ

t

Q

t

t

t. 0

t

TNACK

ME3k4

TRLC/MAC Block

E3R4

E3R45 Pr3C@I# R411-PB2a

M

k5 1

kPk-1B 1 1-Pr3C@I# R4

TLLC RLC/MAC 5 N #E3R4#TRLC/MAC Block 1 TNACK#E3k4

LLC Timer5 1Q 1 12TLLC RLC/MAC 1 t

M

Q

TLLC Service Time

TLLC RLC/MAC

M

M

Aq R

R

CI

CI

=− + ≤( )

≤( )1 Pr

Pr

MA

P

N

=−( )

+ln

ln B

11

1

CDF1M 25 1-Pr1C@I# R21 Pr1C@I# R211-PMB2N 5 q

q

CDF1M 25 a

M

k5 1

Pr1k2

kM

k


measurements taken at the RLC/MAC queue.

This type of ARQ mechanism belongs to the fam-

ily of channel-dependent algorithms.13,14 The time

diversity introduced by this ARQ scheme increases MS

battery lifetime by deferring an LLC frame transmis-

sion over a bad radio channel to a later time when the

radio channel is likely to be in a good state. This

reduces the number of LLC retransmissions at the

expense of a slight increase in transfer delay due to the

additional delay introduced by in (16). Henceforth,

we consider that is large enough such that two con-

secutive LLC retransmissions are uncorrelated. Using

this statistical independence of the LLC retransmis-

sions, we next derive the LLC frame error rate.

The probability that after RLC/MAC retransmis-

sions there are still RLC/MAC blocks in error belong-

ing to the same LLC frame of size is equal to

(19)

where the approximation is valid for . Thus,

is the probability of RLC/MAC failure, which leads

to an LLC retransmission of the whole LLC frame;

then the RLC/MAC process is repeated. In other

words, is the LLC frame error rate seen at the LLC

layer. Therefore, the LLC peer-to-peer link has an

error probability of .

We can see from (19) that the reliability of the

LLC peer-to-peer link increases as increases for a

fixed LLC frame size . When , then the LLC

frame error rate decreases exponentially as

increases, as shown in the second part of (19).

As an example, we set , , and

in (19) to obtain . If we now assume

that an MS sends 5 e-mails/hour (1 e-mail =

1 LLC frame = 50 RLC/MAC blocks) and there are

60 GPRS MSs active in one cell, then the average

number of LLC retransmissions per hour is

. Thus, 10% of the LLC

frames are retransmitted in this example. A much

lower number of LLC retransmissions can be achieved

by implementing a channel-dependent RLC/MAC

protocol.13,14 Once the LLC frame error rate has been

derived in (19), we can calculate the delay and the

throughput per LLC frame.

The LLC transfer delay has been calculated by

Ludwig and Turina12 for the case where the

RLC/MAC layer is capable of an infinite number of

retransmissions and where a uniform radio channel is

assumed (that is, is constant). They calculated the

LLC queuing delay by considering a Poisson arrival

process and, therefore, their result is limited.

Numerical simulations for the LLC transfer delay and

throughput have also been presented elsewhere.7-11

Here, we derive the LLC frame service time for a

finite number of RLC/MAC retransmissions per LLC

frame, , and for a radio channel characterized by a

two-state Markov process (that is, it has good and bad

states). The throughput per LLC frame is also derived.

We explicitly account for the headers of the LLC and

RLC/MAC layers.

For clarity, we recall here that the LLC frame size

is equal to RLC/MAC blocks, the maximum number

of RLC/MAC retransmissions per LLC frame is as

given in (12) or (15), the maximum number of LLC

retransmissions is , and the probability that the radio

channel is in a bad state is equal to .

Then, the average LLC frame service time is

(20)

where and are given in (16) and (19).

Equation (20) comes from the statistical independence

of the LLC retransmissions due to the LLC time-

diversity time-out. When the number of LLC retrans-

missions reaches , then the LLC frame is dropped and

higher layers are in charge of recovery (for example,

the transmission-control protocol [TCP] layer).

There is a tradeoff between the number of LLC

retransmissions and the number of RLC/MAC retrans-

missions. As the number of RLC/MAC retransmissions

per LLC frame, , increases, then the probability of

RLC/MAC failure, , decreases; therefore, fewer LLC

retransmissions are necessary per LLC frame, . Thus,

for large and for a radio channel in a bad state, the

radio channel is heavily loaded with RLC/MAC retrans-

missions. This leads to poor radio efficiency and a waste

of MS battery lifetime. On the other hand, a very small

number of RLC/MAC retransmissions, , gives rise toM

M

L

PM

M

L

PMLLC Timer

+ −( ) −

=∑LLC Timer P kPM M

k

k

L

1 1

2

T P LLC Timer tMLLC Frame Service Time = −( ) −( )1

Pr1C@I# R25 0.3

L

M

N

M

PB

LLC retrans5 5#60#PM 5 30

PM 5 0.1PB 5 0.2

N 5 50M 5 3

M

PMB V 1@NN

M

PM

PM

PM

PMB V 1@N

PM 5 Pr1C@I# R231-11-PMB2N4 > N#PMBPr1C@I# R2

N

M

t

t


many LLC retransmissions (large ), which consequently

leads to heavy network load over the Gb and A-bis inter-

faces. Here Gb refers to the interface between the BSS

and the serving GPRS support node (SGSN), while A-bis

is the interface between the base transceiver station

(BTS) and the base station controller (BSC). This also

increases the LLC frame transfer delay. In counterbal-

ance, the efficiency over the radio interface is increased

due to the inherent time diversity of the LLC retransmis-

sion process (that is, its basis on time-out). We believe

that the expression of given in (12) or (15) strikes a

balance between the number of LLC retransmissions and

the number of RLC/MAC retransmissions.

LLC ThroughputThe throughput seen at the LLC layer can be

defined as follows

.(21)

Equations (20) and (21) enable us to calculate on line

the LLC frame transfer delay and throughput.

We consider that GPRS packet transfer is operated

in the acknowledge mode. Transmission control proto-

col/Internet protocol (TCP/IP) introduces headers of

5 octets each (a 4-to-1 compression algorithm is

applied on a 20-octet TCP/IP header). With reference

to Figure 1, an IP packet is segmented at the subnet-

work dependent convergence protocol (SNDCP) layer,

where a 2-octet header is added to every segment and

the resulting packet is called an SNDCP-PDU

(SN-PDU).4 These SN-PDUs are then passed to the

LLC layer, where they are segmented into LLC frames,

and a 6-octet header is added to each LLC frame. The

LLC frames are passed to the RLC/MAC layer, where

they are again segmented; here, a header of 6 octets is

added to every RLC/MAC block when coding scheme

one (CS-1) is considered. Segmentation at every GPRS

protocol layer is necessary to achieve the residual error

rate required for data transmissions6 (for example, a

residual error rate of for reliability class 2).

Therefore, most of the headers added at any segment

include CRC check bits, among others.

The sizes of the SN-PDU and LLC frames are vari-

able, and only their maximum values are standard-

ized.3,4 In a GPRS implementation, the segmentation

size at any layer is a function of the error rate at the

lower layers, among which the error rate over the

radio interface is of the foremost importance. As

pointed out by Zorzi and Rao,14 a multi-layer protocol

stack should be designed by taking into account the

interactions between the protocol layers and the radio

channel, where most of the errors occur.

In this analysis, the focus is on two protocols—

namely, LLC and RLC/MAC—and we neglect the

overhead due to IP and SNDCP. Therefore, we assume

that the information data field of an IP/SN-PDU packet

is much larger than their headers. Thus, we consider

that SN-PDUs coming to the LLC layer contain only

information bits. The SN-PDUs received at the LLC

layer from the SNDCP layer are segmented into LLC

frames of variable size from 20 to 1500 octets.

We consider that an LLC frame has informa-

tion octets and header octets. An LLC frame

is passed to the RLC/MAC layer, where it is segmented

into RLC/MAC blocks of octets of data

information and header octets for CS-1.

Then, the throughput at the LLC layer is

(22)

with given in (17), where

(23)

and the brackets [ ] denote the closest integer less than

or equal to the term inside the brackets. The

RLC/MAC header, , was already taken into

account in (16), (19), and (21) because the

data-information octets are sent in an RLC/MAC

block, , that also carries the RLC/MAC

header.1

In Figure 4, we plot the throughput at the LLC

layer—as given in (22)—as a function of the LLC

frame size in RLC/MAC blocks—as given in (23)—for

different RLC/MAC block error rates and probabilities.

The simulations in Figure 4 are obtained under the

assumptions that the maximum number of LLC

retransmissions per LLC frame, , is equal to 10 and

that the time shift, , used in the from (16)LLC Timert

L

TRLC/MAC Block

nRLC/MAC

HRLC/MAC

N 5 cnLLC 1 HLLC

nRLC/MACd

TLLC Frame Service Time

ThroughputLLC 5nLLC


HRLC/MAC 5 6

nRLC/MAC 5 20

HLLC 5 6

nLLC

10-6

ThroughputLLC 5information data per LLC frame


M

L


is equal to 50 ms. For simplicity, we also consider that

there is no LLC frame queuing at the RLC/MAC

layer—that is, in (16). In Figure 4a,

is kept constant and we change .

In Figure 4b, is kept constant and we change

. We see from Figure 4a that, by decreasing

from 0.5 to 0.2, the LLC throughput increases by

almost 2 kb/s. It is worth pointing out that the maxi-

mum achievable throughput for CS-1 is equal to

9.05 kb/s.1,2 Thus, for the case where ,

, and for an average LLC frame size

(about 20 RLC/MAC blocks), we can achieve 77%

bandwidth efficiency at the LLC layer. Notably, this

throughput efficiency is achieved while the MS battery

lifetime is increased because of the time diversity

introduced at the LLC layer.

In Figure 4b, the LLC throughput also increases as

(the probability that the radio channel

is in a bad state) decreases. The jumps of the LLC

throughput in Figure 4b along the vertical dotted lines

come from the variation of the maximum number of

the RLC/MAC retransmissions per LLC frame, —see

(12) or (15)—also plotted on the same figure.

According to (19), as increases, the probability of an

LLC retransmission, , decreases exponentially. Thus,

the LLC peer-to-peer link becomes more reliable and

fewer LLC retransmissions are required at the LLC

layer to recover from errors. Fewer LLC retransmis-

sions lead to shorter LLC service time and, therefore,

higher throughput. The compromise is that the MAC

layer should increase its number of retransmissions per

LLC frame, reducing the bandwidth efficiency over the

radio channel. For a constant value of , the LLC

throughput decreases slightly as increases because

the LLC retransmission probability, , increases as the

LLC frame size, , increases—see (19).

In Figure 4b, we also plot LLC throughput for

the case when the LLC layer does not take into

account the radio channel state—that is, when

for non-channel-dependent LLC

protocol. As we see, the throughput in this case is

much lower than that of the channel-dependent LLC

Pr1C@I# R2; Pr 5 1

N

PM

N

M

PM

M

M

Pr ; Pr1C@I# R2

Pr1C@I# R25 0.3

PB 5 0.2

PB

Pr1C@I# R2

PB 5 0.2

PBPr1C@I# R25 0.3

Q 5 0

LLC – Logical link controlM – Maximum number of RLC/MAC retransmissions per LLC frameMAC – Medium access control

PB – Block error ratePr – Probability that the radio channel is in a “bad” stateRLC – Radio link control

0

1

2

3

4

5

6

7

8

2 22 42 62 82

LLC frame size (N)

0

1

2

3

4

5

6

7

8

2 22 42 62 82LLC frame size (N)

PB = 0.2PB = 0.5

LLC

th

rou

gh

pu

t (k

b/s

)

LLC

th

rou

gh

pu

t (k

b/s

)

Pr = 0.3Pr = 0.6

(a) (b)

Pr = 1M

Figure 4.Throughput at the LLC layer versus LLC frame size.


protocol. It is worth pointing out that a similar channel-

dependent ARQ mechanism can be implemented at

the RLC/MAC layer that will substantially increase the

radio efficiency for delay-tolerant traffic.

ConclusionsIn this paper, we have analyzed the

LLC-RLC/MAC protocol stack for GPRS. The interac-

tion between the two ARQ mechanisms at the two

protocol layers has been investigated. A channel-

dependent ARQ has been proposed at the LLC layer

that reduces the number of LLC retransmissions per

LLC frame at the expense of slight degradation in the

delay per LLC frame. This channel-aware ARQ mecha-

nism can substantially increase MS battery lifetime.

The proposed channel-dependent ARQ mechanism

implemented at the LLC in this paper can be easily

introduced at the RLC/MAC as well. This will substan-

tially reduce the number of LLC retransmissions, and

may even remove the need for an ARQ at the LLC

layer because the channel-related control is moved

entirely to the RLC/MAC layer.

The tradeoff between the numbers of LLC and

RLC/MAC retransmissions has been addressed. In par-

ticular, it has been shown that, for delay-tolerant traf-

fic, an optimal number of RLC/MAC retransmissions

leads to better radio-channel bandwidth efficiency and

better MS energy management—especially in a bad

radio environment.

At the RLC/MAC layer, a closed-form expression

for the probability of RLC/MAC retransmission per

LLC frame has been derived. This has allowed us to

analytically compute the most likely RLC/MAC

retransmission number per LLC frame. This formula

can be used for on-line estimation of the throughput

and the transfer delay per LLC frame. In this way, QoS

parameters can be more accurately characterized,

improving GPRS network performance.

AppendixIn this appendix, we show that the formulas

derived in a recent paper by Ludwig and Turina12 for

the probability of having negative acknowledgments

per LLC frame of a size equal to RLC/MAC blocks

are identical with those derived here in (2) through (4).

Ludwig and Turina used an iterative method to com-

pute these probabilities as follows:

(A1)

(A2)

where is the number of NACKs sent per LLC

frame and is the RLC/MAC block error rate. In

(A1) and (A2), we denote by the proba-

bility of receiving NACKs for an LLC frame having a

size equal to RLC/MAC blocks.

We prove now that the iterative process given by

(A1) and (A2) leads to a closed-form expression for

the probability , namely,

(A3)

where is given in (4). We use the method

of induction to prove (A3), knowing that (A1) and

(A2) are true for any integer . Setting in

(A3) yields

, (A4)

which is identical with (A1) and therefore true.

Following the induction method, we assume now that

given by (A3) is true, and we prove that

.(A5)

We rewrite (A2) for in the form

, (A6)=

−( ) ( ) =[ ]−

=∑ N

kP P M k mk

N k

k

N

B B Pr11

Pr M N m( ) = +[ ]1

1m 1 12S m

Pr B BM N m P PmN

mN

( ) = +[ ] = −( ) − −( )+ +1 1 12 1

Pr3M 1N25 m 4

Pr3M 1N25 045 11-PB2N

m 5 0m $ 0

PrC1m 1 12

Pr Pr ( )B B CM N m P P mmN

mN

( ) =[ ] = −( ) − −( ) ≡ ++1 1 11

Pr3M 1N25 m 4

k

m

Pr3M 1k25 m 4

PB

M 1N2

Pr PrB BM N mN

KP P M k mk

N k

k

N

( ) =[ ] =

−( ) ( ) = −[ ]−

=∑ 1 1

1

Pr BM N PN( ) =[ ] = −( )0 1

N

m


which, by using (A3) for , yields

. (A7)

We can rewrite in the form

, (A8)

which, when inserted back into (A7), yields

. (A9)

After some rearrangements, (A9) becomes

(A10)

and, finally,

(Q.E.D.). (A11)

References1. European Telecommunications Standards

Institute, “Digital Cellular TelecommunicationsSystem (Phase 2+); General Packet RadioService (GPRS); Mobile Station (MS) – BaseStation System (BSS) Interface; Radio LinkControl/Medium Access Control (RLC/MAC)Protocol,” EN 301 349 (GSM 04.60), Ver. 6.3.0,May 1999, http://www.etsi.org

2. European Telecommunications StandardsInstitute, “Digital Cellular TelecommunicationsSystem (Phase 2+); General Packet RadioService (GPRS); Overall Description of theGPRS Radio Interface; Stage 2,” TS 101 350(GSM 03.64), Ver. 6.1.0, Nov. 1998,http://www.etsi.org

3. European Telecommunications Standards Insti-tute, “Digital Cellular Telecommunications Sys-tem (Phase 2+); General Packet Radio Service(GPRS); Mobile Station (MS) – Serving GPRSSupport Node (MS-SGSN) Logical Link Control(LLC) Layer Specification,” TS 101 351 (GSM 04.64),Ver. 6.3.0, April 1999, http://www.etsi.org

4. European Telecommunications StandardsInstitute, “Digital Cellular TelecommunicationsSystem (Phase 2+); General Packet RadioService (GPRS); Mobile Station (MS) – ServingGPRS Support Node (MS-SGSN); SubnetworkDependent Convergence Protocol (SNDCP),”TS 101 297 (GSM 04.65), Ver. 6.3.0,April 1999, http://www.etsi.org

5. European Telecommunications StandardsInstitute, “Digital Cellular TelecommunicationsSystem (Phase 2+); Channel Coding,”EN 300 909 (GSM 05.03), Ver. 6.2.0, May 1999,http://www.etsi.org

6. European Telecommunications StandardsInstitute, “Digital Cellular TelecommunicationsSystem (Phase 2+); General Packet RadioService (GPRS); Service Description; Stage 1,”EN 301 113 (GSM 02.60), Ver. 6.2.0, May 1999,http://www.etsi.org

7. J. Cai and D. J. Goodman, “General PacketRadio Service in GSM,” IEEE Commun. Mag.,Vol. 35, No. 10, Oct. 1997, pp. 122–131.

8. G. Brasche and B. Walke, “Concepts, Services,and Protocols of the New GSM Phase 2+General Packet Radio Service,” IEEE Commun.Mag., Vol. 35, No. 8, Aug. 1997, pp. 94–104.

9. D. Turina, P. Beming, E. Schoster andA. Andersson, “A Proposal for Multi-Slot MACLayer Operation for Packet Data Channel inGSM,” Proc. of ICUPC—5th Internat. Conf. onUniversal Personal Commun., Vol. 2, IEEE,Oct. 1996, pp. 572–576.

10. G. Brasche, “A Frame Mode Bearer ServiceProposed for the Cellular Radio Network GSM,”Proc. of 46th Vehicular Technology Conf., Vol. 1,IEEE, April–May 1996, pp. 512–516.

11. S. Hoff, M. Meyer, and A. Schieder, “APerformance Evaluation of Internet Access viathe General Packet Radio Service of GSM,”VTC-98, Proc. of 48th Vehicular Technology Conf.,Vol. 3, IEEE, May 1998, pp. 1760–1764.

12. R. Ludwig and D. Turina, “Link Layer Analysisof the General Packet Radio Service for GSM,”Proc. of ICUPC—6th Internat. Conf. on UniversalPersonal Commun., Vol. 2, IEEE, Oct. 1997,pp. 525–530.

13. M. Zorzi, R. R. Rao, and L. B. Milstein, “ARQError Control for Fading Mobile Channels,”

≡ +PrC( )m 2

Pr B BM N m P PmN

mN

( ) = +[ ] = −( ) − −( )+ +1 1 12 1

− + + + +( )

1 2P P PmN

B B B...

= −( ) + + + +( )

+1 1 2 1P P P PB

Nm

N

B B B...

Pr M N m( ) = +[ ]1

−

+ + +( )

=∑ N

kP P P M

k

k

N

B B B2

0

...

= −( )

+ + +( )

+

=∑1 2 1

0

PN

kP P P

Nm

k

k

N

B B B B...

Pr M N m( ) = +[ ]1

11-PmB2k 5 11-PB2

k 11 1 PB 1 P2B 1 ... 1 Pm-1B 2k

11-PmB2k

=

−( ) −( ) − −( )

−+

=∑ N

kP P P Pk

N km

km

k

k

N

B B B B1 1 11

1

Pr M N m( ) = +[ ]1

Pr3M 1k25 m 4


IEEE Trans. on Vehicular Technology, Vol. 46,No. 2, May 1997, pp. 445–455.

14. M. Zorzi and R. R. Rao, “The Role of ErrorCorrelations in the Design of Protocols forPacket Switched Services,” 35th Ann. AllertonConf. on Commun., Control, and Computing,Univ. of Illinois, Urbana, Sept.–Oct. 1997,pp. 1–9.

15. E. N. Gilbert, “Capacity of a Burst-NoiseChannel,” Bell System Tech. J., Vol. 39,Sept. 1960, pp. 1253–1266.

16. U.S. National Bureau of Standards, “Handbook ofMathematical Functions with Formulas, Graphs, andMathematical Tables,” 9th ed., edited byM. Abramowitz and I. A. Stegun, DoverPublications, New York, 1972.

(Manuscript approved October 1999)

CRISTIAN DEMETRESCU is a member of technical staffin the Global Wireless Systems ResearchDepartment of Bell Labs in Swindon,Wiltshire, United Kingdom. Prior to joiningBell Labs, he received a B.Sc. in electricalengineering from the University of Craiova

in Romania and a Ph.D. in electrical engineering fromthe University of Birmingham in the United Kingdom.He is currently working on real-time services (such asvoice) over the Enhanced General Packet Radio Service(EGPRS) and is team coordinator for packet voice overGPRS prototyping. ◆

llc-mac analysis of general packet radio service in gsm

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