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A Stochastic Game Analysis of the Binary Exponential Backoff Algorithm with Multi-Power Diversity and Transmission Cost

A Stochastic Game Analysis of the Binary Exponential Backoff Algorithm with Multi-Power Diversity and Transmission Cost
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  J Math Model AlgorDOI 10.1007/s10852-012-9190-8 A Stochastic Game Analysis of the Binary ExponentialBackoff Algorithm with Multi-Power Diversityand Transmission Cost Abdelillah Karouit · Essaid Sabir · Fernando Ramirez-Mireles · Luis Orozco Barbosa · Abdelkrim Haqiq Received: 6 October 2011 / Accepted: 12 April 2012© Springer Science+Business Media B.V. 2012 Abstract  In this paper, we present a game analysis of the Binary ExponentialBackoff (BEB), a popular bandwidth allocation mechanism used by a large numberof distributed wireless technologies. A Markov chain analysis is used to obtainequilibrium retransmission probabilities and throughput. Numerical results showthat when the arrival probability increases, the behavior of mobile stations MSsbecome more and more aggressive resulting in a global deterioration of the systemthroughput. We then consider a non-cooperative game framework to study theoperation and evaluate the performance of the BEB algorithm when a group of MSs competing with each other to gain access to the wireless channel. We focusour attention to the case when an MS acts selfishly by attempting to gain accessto the channel using a higher retransmission probability as a means to increase itsown throughput. As a means to improve the system performance, we further explorethe use of two transmission mechanisms and policies. First, we introduce the use of  A. Karouit  ·  A. HaqiqL-IR2M, Faculty of Sciences and Techniques, Hassan 1st University, Casablanca, B.P. 577,Settat, MoroccoA. Karouite-mail: akarouit@gmail.comA. Haqiqe-mail: ahaqiq@gmail.comE. SabirLIA/CERI University of Avignon, Agroparc, BP 1228, 84911, Avignon, Francee-mail: essaid.sabir@univ-avignon.frF. Ramirez-MirelesEngineering Division, ITAM, Mexico City, Mexicoe-mail: ramirezm@ieee.orgL. Orozco Barbosa ( B )University of Castilla La Mancha (UCLM), 02071 Albacete, Spaine-mail:  J Math Model Algor multiple power levels (MPLs) for the data transmission. The use of multiple powerlevelsresultsonacaptureeffectallowingthereceivertoproperlydecodethemessageeven in the presence of a collision. Under the proposed scheme, named MPL-BEB,the effect of the aggressive behavior, higher transmission probabilities, is diminishedsince the power level is chosen randomly and independently by each and everystation.Second,weintroduceadisutilitypolicyforpowerconsumption.Theresultingmechanism, named MPL-BEB with costs, is of prime interest in wireless networkscomposed of battery-powered nodes. Under this scheme aggressive behavior isdiscouraged since each retransmission translates into the depletion of the energystored in the battery. Via price of anarchy, our results identify a behavior similar tothe well-know prisoner’s dilemma. A non-efficiency of Nash equilibrium is observedfor all schemes (BEB, MPL-BEB, MPL-BEB with costs) under heavy traffic with anotable outperformance of MPL-BEB with costs over both MPL-BEB and BEB. Keywords  Random access · Capture effect · Nash equilibrium · Price of anarchy · Stochastic game · BEB algorithm with multi-power diversity · Pricing 1 Introduction With its potential synergy in analyzing actors/players behaviors and predicting theoutcome of a conflict situation (called “game”), game theory is the ultimate tool toadopt when studying decentralized systems. Under rationality of actors/players, themost common solution concept is called the “Nash Equilibrium”.A Nash equilibrium point is a strategy profile where no player has any incentive todeviateunilaterally.Inthepastfewyears,therehasbeenanincreasinginterestonthebenefits of applying the principles of game theory in order to better understand andplan the operation of telecommunication systems. Recently, the effects of the selfishbehavior of mobile stations (MSs) has been widely analyzed using game theory inorder to better understand the operation of the network, MAC (Medium AccessContro) and physical mechanisms employed by such wireless networks.For instance many works on power control, random access, routing games havebeen reported in the literature, see [4, 6, 8, 10, 13] and [12]. In particular, in this work we undertake the study of the Binary Exponential Backoff mechanism (BEB),a widely used channel access conflict resolution mechanism. The scenario of interestconsists of a distributed network where a group of MSs make use of the BEBmechanism in order to communicate with a central entity, the Access Point (AP). Tosolve the conflicts arising when more than one MS attempt to simultaneously accessthe channel, the MSs implement the BEB protocol. Our study is formulated usingthe principle of game theory. The game scenario under study is made up of threeelements: (1) The decision makers are the mobile stations, (2) The individual actionis either to “transmit” or to “wait”, the retransmission probability is therefore thestrategy. The resulting strategy profile is given by the network load (outcome of thegame), and (3) The utility function of each MS is its own throughput (or alternativelyminus its delay). Clearly, the payoff of each station depends not only on its owndecision, but also on the actions of the other (adversarial) stations. Although ourscenario does have a centralized entity, this entity does not have the full picture of thenetworkconditionscreatedbytheindividualdecisionsoftheMS.Themainaimis  J Math Model Algor therefore to predict what might or should happen when an aggressive mobile stationcompeting with other MSs acts selfishly using a higher retransmission probability inorder to increase its own throughput. In this case, a performance collapse is thenpredicted for collision-channel systems using Aloha or the BEB algorithm, see [2, 7]. In the BEB mechanism, when an MS tends to contend for a transmissionopportunity, it enters a backoff procedure [5]. Whenever the channel is sensed busy, the MS defers the transmission of its message and waits for an idle channelopportunity. In addition and in order to reduce the collision probability due to twoor more simultaneous transmissions, the BEB procedure employs a slotted binaryexponential backoff where an MS is forced to wait for a random number of timeslots, also called “backoff counter” before attempting to access the channel onceagain. The backoff time is uniformly chosen from the interval  [ 0 , W   −  1 ] , where  W   isthe current contention window that mainly depends on the number of experiencedcollisions. The contention window is dynamic and given by  W  k  =  2 k W  0 , where  k represents the backoff stage, i.e., it represents the current retransmission attemptnumber for the message being transmitted, and  W  0  is the initial contention window.The window is not doubled when the maximum backoff stage is reached. Thebackoff counter is decremented by one whenever the channel is sensed idle, whileit freezes if the channel is sensed busy. Finally, when the message is transmitted,the MS has to wait for an acknowledgement, either implicit ACK or explicit ACKdepending on the particular network technology being employed. If no notificationis obtained in a certain number of subsequent slots, the message is considered lostand a retransmission has to be re-scheduled. When the number of retransmissionsexpires, the concerned mobile station discards the message. In the traditional BEB,although the terminals compete, they do not do so on the basis of a decision-makingprocess. This is to say, the MSs are not given an individual retransmission probability,but rather they all use a predefined set of parameters or a parameter given by acentral entity. In this case finding the most convenient retransmission probabilityensuring the correct or even the optimal operation of the overall network may beonly carried out by a single decision maker: the central entity, commonly referredin the literature as the Base Station (BS). In this work, we develop a stochasticgame-based framework to model and evaluate BEB while exploiting power diversity.Indeed, previous works have verified that the performance of a network makinguse of such mechanisms can be improved when: (1) The MSs use power diversity,i.e., before starting transmission each and every MS picks randomly a power levelamong  N   different available levels. This power diversity produces a “capture effect”in which even when two or more messages collide, i.e., one of them can be decodedsuccessfully with a certain probability [2, 13], and (2) The value of the initial window W  0 , directly related with the retransmission probability, can be optimized when theMSs cooperate with each other (team theory) [9], or even in the case when the MSsact selfishly (game theory) [10]. Previous works have combined team theory andpower diversity for Aloha [2, 7] and use game theory with power diversity for Aloha [7]. Indeed, it has been verified that the system performance can be improved whenassociating a cost to each transmission attempt. In fact, it has been shown that theuse of such pricing scheme can be used to match the equilibrium throughput of theAloha protocol with the optimal team throughput [1].In this work, we develop a stochastic game-based framework with transmissionand retransmission costs and evaluate BEB under battery constraints. To the best of   J Math Model Algor our knowledge, this work proposes for the first time in the literature the implemen-tation of power diversity and transmission costs in a system operating using the BEBto increase throughput and reduce delay in a decentralized context (selfish mobilestations).Therestofthepaperisorganizedasfollows.Wefirstdescribethedesignprinciplesof power diversity as related to its use in conjunction with the BEB mechanismin Section 2. In this way, we set the basis to understand the existing interaction between the physical and the MAC layers of the protocol stack following a cross-layer approach. In Section 3, we formulate the stochastic game approach for both mechanisms. For completeness purposes, Section 4 introduces the equilibrium analy-sis principles used in the evaluation of the mechanisms under study, i.e., the BEBand MPL-BEB mechanisms. We then derive the performance metrics of interestin Section 5. We provide a numerical evaluation of both mechanisms in Section 6. Finally, our concluding remarks are drawn in Section 7. 2 Cross-Layer Modelling and Main Notations Weconsiderawirelessmultipleaccesssystemcomposedofonecentralreceiver(basestation BS) and  m  geographically dispersed mobile stations communicating with theBS. There is no central control and consider that the mobile stations make use of the BEB mechanism for overcoming all channel access conflicts. Time is divided intomultiple equal and synchronized slots. Transmission feedback (success or collision)is received at the end of the current slot. As mentioned before, the use of multiplepower levels, also referred as power diversity, is to be considered in conjunction withtheBEBmechanism.Throughoutthispaper,wewillrefertotheresultingmechanismas the Multiple Power Level BEB mechanism or simply MPL-BEB. The use of suchfacility will give place to a capture effect. Due to this effect, a receiver may be ableto decode a message even in the presence of a collision. In fact, the unsuccessfulconcurrent messages are lost and treated as interference.In this MAC/Physical cross-layer design an MS contending for a messagetransmission, randomly chooses a power level  T   j   among  N   available levels  T   ={ T  1 , T  2 ,..., T  N  } . The power levels random selection follows the probability vector  X   = [  x 1 ,  x 2 ,...,  x N  ] , where the  j  - th  entry  x  j   is the probability of choosing powerlevel the power level  T   j  . We consider a general capture model where a messagetransmitted by an MS  i  is received successfully when and only when its Signal toInterference plus Noise ratio (SINR) exceeds some given threshold   th . Let  σ  2 bethe variability of the thermal noise and denote by  S  = [ S 1 , S 2 , ···  , S m ]  the vector of selected power level at the beginning of the current slot. Note that components of   S are selected from the vector  T  . The received power on the BS can be related to thetransmitted power by the propagation relation  h i  ·  S i , where  h i  is the channel gainexperienced by the base station when receiving a message transmitted by MS  i . Notethat  h i  does not depend on the value of using power level  S i . Thus, the instantaneousSINR of user  i  transmitting at power level  S i  experienced by the receiver is  i ( S )  = h i  ·  S im  k = 1 , k = i h k  ·  S k  · 1 k  +  σ  2 ,  (1)  J Math Model Algor where  1 k  is an indicator function of the event that at the current slot, MS  k  transmitsits message. We denote by  A  s  ,  s  ≥  2 , the probability of a successful transmissionamong  s  simultaneous attempts, i.e., transmitted during the same slot. Let us denoteby  a  si  the probability that transmission of some tagged MS  i  is successful while  s −  1 other mobile stations simultaneously attempt to transmit.  a  si  can be derived using thefollowing events decomposition  a  si  = N    j  = 2 P  ( A  j i ,  s  B  j i ,  s  C  j i ,  s )  where– A  j i ,  s  is the event “ Mobile station i attempts transmission with power level T   j  ”.– B  j i ,  s  is the event “ Others  −  1 mobilestationstransmitwithpowerlevelslowerthanT   j  ”.– C  j i ,  s  is the event “  Instantaneous SINR of MS i is higher than the target SINR   th ”.Since all mobile stations are symmetric and are assumed to experience the samechannel gain, i.e.,  h i  =  h ,  i  =  1 ,..., m . Then  A  s  =  s . a  si , it follows that  A  s =  s N  − 2  l  = 0  s − 1   s 1 = 0 ···  s − 1   s N  − l  − 1 = 0  x  s 1 1  ·  x  s 2 2  ···  x  s N  − l  N  − l   · u  T  N  − l N  − l  − 1  r  = 1 T  r   s r   +  σ  2 h −   th  · δ   s − 1 − N  − l  − 1  r  = 1  s r   , (2)with,  A 0  =  0  and  A 1  =  1 . In order to get a successful transmission, we need to take  s N  − l   =  1 . Moreover,  T  N  − l   is the power level chosen by the MS whose transmissionmay potentially succeed, i.e., the one corresponding to the highest power selectedin the current slot. Whereas  s r   denotes the number of mobile stations that havechosen the power level  T  r   in the current slot.  δ( t  )  is the Kronecker’s delta functionand  u ( t  )  is the Heaviside function (unit step function) are given by the followingexpressions δ( t  )  =  1 ,  if   t   =  0 , 0 ,  else .  and  u ( t  )  =  1 ,  if   t   ≥  0 , 0 ,  else .  (3)Computing the success probability is a challenging task. The difficulty of formula(2) is to consider one single MS transmitting with power level T and list all thecases where the  s −  1  remaining mobile stations transmit at lower power levels. Thiscorresponds exactly to the set of partitions 1 of the positive integer  k  −  1  consideringall possible permutations. Generating all the partitions of an integer has been widelystudied in the literature and several algorithms have been proposed, e.g., see [3]. The computational complexity of such algorithms is very expensive and may take longtime to compute the set of all partitions and their permutations. Fortunately, in ourmodel the success probability depends on none of the following: the instantaneousstate of the system  n ; the arrival probability  λ ; and, the retransmission probability  q . 1 A partition of a positive integer  η  is a way of writing  η  as a sum of positive integers.
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