Optimization Performance and Pricing in the IEEE802.16 Networks

Abdelillah Karouit and Abdelkrim HaqiqMathematics and Computer Science Department

Science and Technical FacultyHassan 1 university, Settat Morocco

e-NGN research group, Africa and Middle EastEmail: {arkaouit ; ahaqiq}@gmail.com

Luis Orozco BarbosaAlbacete Research Institute of InformaticsUniversity of Castilla La Mancha (UCLM)

02071 Albacete, SpainEmail: luis.orozco@uclm.es

Abstract—In this paper, we study the collision resolution pro-tocol of the IEEE 802.16 standard for metropolitan broadbandwireless access systems. The standard specifies the use of arandom access scheme based on the slotted binary exponentialbackoff (BEB) algorithm during the bandwidth request phase. Weanalyze the distributed choice of the retransmission probabilitiesused by the BEB protocol. We follow a cooperative team problemapproach towards the analysis of the throughput and the delayduring the bandwidth reservation process. A Markov chainanalysis is used to obtain the optimal retransmission probabilities,the optimal value of the initial backoff window size, Wmin, thethroughput and the expected delays experienced by a subscriberstation (SS) for placing its bandwidth requests. We then inves-tigate the impact of adding retransmission costs (which mayrepresent the disutility for power consumption during bandwidthrequest phase), and show how this pricing can penalizes thethroughput of a request transmission.

Keywords:IEEE 802.16, Binary exponential backoff, Teamproblem, Markov chain, Pricing.

I. INTRODCUTIONThe IEEE 802.16 WiMaX standard [1] is supposed to

play an important role in rapid and ubiquitous adoptionof broadband wireless access systems worldwide. In thecentralized point-to-multipoint architecture of WiMaX,subscriber stations (SSs) share the uplink to a base station(BS) on demand basis. The BS receives the requests fromthe SSs and grants them the transmission opportunities intime slots taking into account the QoS requirements of eachand every SS and the available channel resources. Two mainmethods to provide transmission opportunities are suggested:centralized polling and contention-based random access.According to the IEEE 802.16, the latter may be implementedin two different ways. The first and most used solution isto give all stations an opportunity to access all availablecontention slots; another typical solution is to group somestations together and assign disjoint subsets of all contentionslots to each group. In this work, we focus our analysis tothe former. Previous results have been obtained to evaluate itsperformance as well as to propose to vendors some efficientways to use different available tools and methods.

A large number of research papers have focused on theperformance evaluation of the various IEEE 802.16 features

[3]-[4]-[5] . In particular, the bandwidth requests transmissionby a subscriber to reserve a portion of the channel resources isfrequently addressed. A detailed description of the reservationtechniques is known from the fundamental work in [6]. Thestandard allows a random multiple access (RMA) scheme atthe reservation stage and implements the truncated binaryexponential backoff (BEB) algorithm for the purposes of thecollision resolution. The asymptotic behavior of the BEBalgorithm has been widely addressed in the literature. In[7] it was shown that the BEB algorithm is unstable in theinfinitely-many subscriber stations case. By contrast, [8]showsthat the BEB is stable for any finite number of subscribesstations, even if it is extremely large, and sufficiently lowinput rate. These seemingly controversial results demonstratethe two alternative approaches to the analysis of an RMAalgorithm [9]. The former is the infinite population model,which studies the ultimate performance characteristics of anRMA algorithm. The latter is the finite population model thataddresses the limits of the practical algorithm operation. Anexhaustive description of both models may be found in [10]and [11].Within the framework of IEEE 802.16, the BEB algorithmwas first investigated in [12] under a ”saturation” conditionassumption by using an analytical approach similar to [14].Following a fully cooperative control approach. We aim tofind the optimal strategy of retransmission probabilities forThe BEB algorithm as to achieve the maximum throughput(channel’s bandwidth utilization) or the minimum delayduring the reservation process.In the following, the paper is organized as follows: SectionII overviews the IEEE 802.16 MAC protocol and associatedrequest mechanisms. We then briefly describe the BEBalgorithm in Section III. In Section IV, we formulate andsolve the team problem. We then provide some numericalresults in Section V. The model with pricing is thenintroduced in Section VI and is investigated numerically inSection VII.Finally we draw our conclusions in Section VI.

978-1-61284-732-0/11/$26.00 ©2010 IEEE

II. IEEE 802.16 MAC PROTOCOL

A. FRAME STRUCTURE

In the following description, we should consider a networkconfigured in a point-to-multipoint (PMP) architecture, i.e.,one BS and multiple SSs whose channel access is managed bythe BS. Transmissions between the BS and SSs are realized infixed-sized frames by means of a time division multiple access(TDMA) / time division duplexing (TDD) mode of operation.The frame structure in the mode TDD is shown in Fig 1comprises a downlink sub-frame for transmission from the BSto the SSs and an uplink sub-frame for transmissions in thereverse direction. The Tx/Rx transition gap (TTG) and Rx/Txtransition gap (RTG) is inserted between the sub-frames allow-ing the stations to turn around from reception to transmissionand vice versa. In the downlink sub-frame the Downlink MAP(DL-MAP) and Uplink MAP (UL-MAP) messages are trans-mitted by the BS, which comprise the bandwidth allocationsfor data transmission in both downlink and uplink directions,respectively. Another important management message, whichis related to UL-MAP, is called an Uplink Channel Descriptor(UCD). It can be periodically transmitted in the downlink sub-frame. The values of the minimum backoff window, Wmin,and the maximum backoff window, Wmax, are defined in thismessage. The proper setting of these two parameters plays animportant role on the performance of the collision resolutionalgorithm used by the standard. Regarding the uplink sub-frame, this one contains a transmission opportunities field usedfor conveying the bandwidth requests (BW-REQ) of the SSsto the BS, i.e., the uplink bandwidth requirements of each SS.

Fig. 1. IEEE 802.16 frame structure in the TDD mode

B. REQUEST MECHANISM IN IEEE 802.16

Each transmission opportunity may be assigned by theBS either to exactly one subscriber station or to a group ofstations. In the first case, a station is provided a so-calledunicast opportunity for issuing its BW-REQ. In other words,the BS polls a SS to allow it to transmit the request in acontention- free manner. In the latter case, the multicastcase, a random access algorithm is used by the group ofSSs to contend for the common transmission opportunitiesand resolve possible collisions. As mentioned earlier, thecontention resolution method, whose inclusion is mandatoryby the standard, is based on a truncated binary exponentialbackoff, with the initial backoff window and the maximumbackoff window controlled by the BS. The details of thecollision resolution algorithm are given in the next section.

The information whether BW-REQ message is successfullytransmitted or distorted (because of collision or noise) is notexplicitly transmitted by the BS in the downlink. The standarddoes not specify how the SSs get to know the outcome oftheir transmission. It might be based on the correspondencebetween the amount of the resources assigned to the givenSS and the amount of the resources it has asked for in thetransmitted BW-REQ message.

III. BINARY EXPONENTIAL BACKOFFIn distributed multiple access, a simple yet effective

random backoff algorithm is widely used to avoid collisions.In particular, the truncated binary exponential backoffalgorithm adjusts the contention window size dynamicallyaccording to the collision intensity. This algorithm works asfollows:

If a request is ready for retransmission at the beginning offrame t at its ith retransmission attempt (i > 0), an SS choosesa number (backoff counter) in the range [0, 2min(m,i)W − 1]uniformly, where W and m are the parameters of the BEBalgorithm, named initial contention window and maximumbackoff stage respectively and i is the number of collisionsthis request suffered from so far.The SS then defers the requestretransmission for the chosen number of slots.

IV. PROBLEM FORMULATIONWe assume that:• The time of BS processing including scheduling is neg-

ligible.• The channel propagation time is negligible.• The transmission channels are error-free.• BW-Req for i-messages arriving during reservation inter-

val can be sent first time in the reservation interval of thenext frame.

• Finite number of SSs having a buffer sufficient to storeexactly one request.

We use a Markovian model based on [16]. We have set themessage size used by all SSs equal to one packet. In termsof bandwidth requesting process, each SS may be either inone of two states: thinking or backlogged. A thinking SShas no message ready for transmission and may generateone during a frame with the probability qa. Once a newmessage is generated, the SS enters the backlogged state whereno new arrivals are possible. This corresponds to the realsystem where after the first arrival a SS starts the contentionprocess and the subsequent arrivals are irrelevant to establishthe sought contention delay. Once the transmission of a BW-Req is successful, the SS enters the thinking state and is ableto generate new messages. In each slot the backlogged SSsattempt to retransmit a BW-Req with probability qr. As In[14] and [15] the probability of retransmission is given by:

Similarly to [15], we consider that the probability qr isgiven as function of backoff window minimal Wmin, qr =

2Wmin + 1

The determination of the above random time can be con-sidered as a stochastic control problem. The informationstructure, however, it is not a classical one: the SSs do not haveknowledge of the full state information. They do not knowthe number of backlogged SSs, nor they do know the numberof messages involved in a collision. We study this controlproblem as a team problem, where there is a common goal toall SSs in the network: to maximize the network throughputor to minimize the message transmission time.We shall restrict in our problem to simple policies in which qrdoes not change over time. Since SSs are symmetric, we shallfurther restrict to finding a symmetric optimal solution, i.e.,all stations attempt to retransmitting with the same probability,denoted from now on by qr.We shall use as the state of the system the number ofbacklogged SSs (or equivalently, of backlogged messages) atthe beginning of a slot, and denote it by n. For any choiceof values qr ∈ [0, 1], the state process is a Markov chainthat contains a single ergodic chain and possibly transientstates as well. The corresponding Markov chain is depictedin figure 2. Define qr to be the vector of retransmission

Fig. 2. Markov chain for the team problem.

probabilities where all SSs whose jth entry is qjr . Let π(qr) be

the corresponding vector of steady state probabilities where itsnth entry, πn(qr), denotes the probabilities of n backloggedSSs. When all entries of qr are the same, say q, we shall writeπ(q) instead of π(qr).Let Qr(i, n) be the probability that i out of the n backloggedSSs retransmit at a given slot. Then

Qr(i, n) =(n

i

)(1− qr)(n−i)qi

r (1)

Let M denote the number of SSs in the network. The proba-bility that i unbacklogged SSs transmits a new message in agiven slot, Qa(i, n) is then given by:

Qa(i, n) =(M − ni

)(1− qa)(M−n−i)qi

a (2)

The transition probabilities of the Markov chain are given by

P(n,n+k)(qr) =

Qr(1, n)Qa(k + 1, n− 1)+(1−Qr(1, n))Qa(k, n) 1 ≤ k ≤M − n(1−Qr(1, n))Qa(0, n)+Qr(1, n)Qa(1, n− 1) k = 0Qr(1, n)Qa(0, n− 1) k = −10 Otherwise

The throughput, defined as the sample average of the numberof messages that are successfully Transmitted, is given by:

thp(qr) =M∑

n=0

πn(q)Qr(1, n) (3)

The team problem is therefore given as the solution of theoptimization:

maxqr

thp(qr) s.c

π(q) = π(qr)P (qr)πn(qr) ≥ 0, n = 0, ....,M∑M

n=0 πn(qr) = 1(4)

A solution to the team problem can be obtained by computingrecursively the steady state probabilities, and thus obtain anexplicit expression for thp(qr) as function of qr.Singularity at qr= 0 : The only point where the Markov chainP does not have a single stationary distribution is at q = 0,where it has two absorbing states: n = M and n = M − 1.All remaining states are transient (for any qa > 0), and theprobability to end in one of the absorbing states, depends onthe initial distribution of the Markov chain. We note that if thestate M − 1 is reached then the throughput is qa. However, ifthe state M is reached, then the throughput equals 0, whichmeans that it is a deadlock state. For qa > 0 and qr = 0, thedeadlock state is reached with positive probability from anyinitial state other than the absorbing state M − 1. We shalltherefore exclude the case of qr = 0 and optimize only on therange ε < q ≤ 1.Existence of a solution The steady state probabilities π(qr)are continuous over 0 < qr ≤ 1. Since this is not a closeinterval, a solution does not exist. However, as we restrict tothe closed interval q ∈ [ε, 1] where ε > 0, an optimal solutionindeed exists. Note also that the limit limqr→0π(qr) existssince π(qr) is a rational function of qr at the neighborhoodof zero. Therefore for any δ > 0, there exists some qr > 0which is δ-optimal. (q∗r > 0 is said to be δ-optimal if it satisfiesthp(q∗r ) ≥ thp(qr)− δ for all qr ∈ [0, 1].In order to calculate the expected delay of the transmittedmessages of the team problem, we should first compute theaverage number of backlogged messages which is given by:

S(qr) =M∑

n=0

πn(qr)n (5)

where the throughput of the number of messages that aresuccessfully transmitted is given by 3.The delay of transmitted messages D, defined as the averagetime slots that a message takes from its source to the receivercan then be obtained by applying Little’s result:

D(qr) = 1 +S(qr)thp(qr)

(6)

The team problem is therefore given as the solution of theoptimization:

minqr

D(qr) s.c

π(qr) = π(qr)P (qr)πn(qr) ≥ 0, n = 0, ....M∑M

n=0 πn(qr) = 1(7)

Existence of a solution The same reasoning as above isvalid for any other objective function and in particularherein towards the minimization of the delay of transmissionmessage ).

V. NUMERICAL RESULTS

In this section we shall obtain the optimal values of theretransmission probabilities and the optimal size of the backoffwindow Wmin. We further investigate the dependence of thethroughput and delay on the arrival probabilities qa and on thenumber of SSs.We consider the team problem in which the performancesto optimize are the throughput and the expected delay oftransmission messages. In Figs. 3 and 4 the expected delayof transmitted messages and the optimal retransmission prob-abilities are plotted versus the arrival probability qa for theteam problem. In Fig. 3 we observe that the delay convergesto infinity when the arrival probability approaches to 1. Forexample for M=20 and qa = 0.9.Fig. 4 shows that the optimal retransmission policy should

Fig. 3. Optimal delay as a function of the arrival probabilities qa forM=2, 4, 10, 20

be reduced as the arrival probability increases or the num-ber of subscribers increases since the system becomes morecongested.

The results show that the optimal retransmission probabili-ties, qr, should be reduced as the arrival probabilities increaseor as the number of SSs is large. In fact, the results showthat for arrival probabilities as low as 0.05 and for M=20, theretransmission probability should be reduced drastically. Fig.

Fig. 4. The optimal retransmission probabilities as a function of thearrival probabilities qa for M = 2, 4, 10, 20

5 shows the throughput for M = 2, 4, 10, 20 for the teamproblem, as a function of the arrival probability qa. Underheavy traffic, the throughput decreases drastically as a functionof the traffic intensity and much faster in the case of a networkconsisting of a larger number of SSs. Fig. 6 shows the optimalwindow backoff Wmin for M = 2, 4, 10, 20 and as a functionof the arrival probabilities. As expected, when the network isexposed to heavier load, larger number of active SSs or higherarrival rates, the window has to be widen in order to reducethe collision probability.

Fig. 5. Optimal throughput as a function of the arrival probabilitiesqa for M=2, 4, 10, 20

It results interesting that under overload conditions, theoptimal throughput converges independently of the number ofactive SSs. For instance, Fig. 7 shows the optimal throughputversus the number of SS for qa = 0.7, 0.9. From the Fig. 7, itis clear that the throughput indeed converges to 0.1438 (resp.0.0510) for qa = 0.7 (resp. qa = 0.9). In fact, under heavytraffic and a large number of SS, the optimal retransmission

probability is seen to be ε.

Fig. 6. The optimal window backoff Wmin as a function of the arrivalprobabilities qa for M = 2, 4, 10, 20

Fig. 7. Optimal throughput as a function of the number of SS forqa = 0.7, 0.9

This explains the convergence of delay to infinity, and theconvergence of throughput to 0 when the arrival probabilityapproach to 1 see Figs. 3 and 5 respectively .

VI. ADDING COSTS FOR RETRANSMISSION

In this section we consider the problem where there is anextra cost θ per each retransmission during bandwidth requestphase. This can represent the disutility for the consumptionof battery energy, which is a scarce resource. For a givensymmetric qr for all SSs, the steady-state retransmission costis θ

∑Mn=0 πn(qr)n where as the cost of messages that enter

the system and are not rejected) is θthp(qr). Thus the new

team problem is

maxqr

{thp(qr)(1− θ)− θqrM∑

n=0

πn(qr)n} (8)

VII. NUMERICAL INVESTIGATION

In this section we obtain the retransmission probabilitieswhich solve the new team problem with the extra transmission

Fig. 8. The optimal throughput as a function of the number of SS forM=4 and θ=0, 0.2, 0.5

costs. We shall investigate the dependence of the solution onthe value θ. In Figs. 8-11 we depict the throughput obtainedat the optimal solution and the optimal retransmission prob-abilities, respectively, as a function of the arrival probability,for the team problem with M=4, 10, for various values of θ.

Fig. 9. The optimal retransmission probability as a function of thenumber of SS for M=4 and θ=0, 0.2, 0.5

We see that both the throughput as well as the retransmissionprobabilities are monotone decreasing in the cost. This canbe expected since retransmissions become more costly withincreasing θ. An interesting feature is that for any fixed

Fig. 10. The optimal throughput as a function of the number of SS forM=10 and θ=0, 0.2, 0.5

θ 6= 0, the retransmission probabilities decrease in the arrivalprobability (which is natural since congestion in the systemincreases as qa increases)

Fig. 11. The optimal retransmission probability as a function of thenumber of SS for M=10 and θ=0, 0.2, 0.5

VIII. CONCLUSION

We have used a team approach for setting up theretransmission probabilities and window size used in thecollision resolution protocol of the IEEE 802.16 standard.Our objective has been to minimize the expected delayand maximize the throughput during the bandwidth requestprocess. Our results show that as the request arrivals increase,the SSs should reduce the retransmission probability bywidening their backoff windows. By adding the cost ofretransmission, we observe much performance degradation inthe throughput during the bandwidth reservation process.Thiscan be expected since retransmissions become more costlyunder overload conditions.

Acknowledgment: This work was partially supportedby the Ministry of Science and Technology of Spain underCONSOLIDER Project CSD2006-46. The first authoracknowledges the support of the University of Castilla LaMancha and Santander Bank.

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