IN many applications, the average delay packets experience. Delay-Minimal Transmission for Average Power Constrained Multi-Access Communications

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1 754 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 9, NO. 9, SEPTEMBER 010 Delay-Miimal Trasmissio for Average Power Costraied Multi-Access Commuicatios Jig Yag, Studet Member, IEEE, ad Seur Ulukus, Member, IEEE Abstract We ivestigate the problem of miimizig the overall trasmissio delay of packets i a multi-access wireless commuicatio system, where the trasmitters have average power costraits. We use a multi-dimesioal Markov chai to model the medium access cotrol layer behavior. The state of the Markov chai represets curret queue legths. Our goal is to miimize the average packet delay through cotrollig the probability of departure at each state, while satisfyig the average power costrait for each queue. We cosider a geeral asymmetric system, where the arrival rates to the queues, chael gais ad average power costraits of the two users are arbitrary. We formulate the problem as a costraied optimizatio problem, ad the trasform it to a liear programmig problem. We aalyze the liear programmig problem, ad develop a procedure by which we determie the optimal solutio aalytically. We show that the optimal policy has a threshold structure: whe the sum of the queue legths is larger tha a threshold, both users should trasmit a packet durig the curret slot; whe the sum of the queue legths is smaller tha a threshold, oly oe of the users, the oe with the loger queue, should trasmit a packet durig the curret slot. We provide umerical examples for both symmetric ad asymmetric settigs. Idex Terms Delay miimizatio, multi-access commuicatio, medium-access cotrol, queue cotrol, power allocatio, cross-layer desig. I. INTRODUCTION IN may applicatios, the average delay packets experiece is a importat quality of service criterio. Therefore, it is importat to allocate the give resources, e.g., average power, eergy, etc., i a way to miimize the average delay packets experiece. Sice power ad eergy are physical layer resources, ad the delay is a medium access cotrol (MAC) layer issue, such resource allocatio problems require close collaboratio of physical ad MAC layers, ad yield cross-layer solutios. Our goal i this paper is to combie iformatio theory ad queueig theory to devise a trasmissio protocol which miimizes the average delay experieced by packets, subject to a average power costrait at each trasmitter. Mauscript received September 8, 008; revised November, 009 ad March 1, 010; accepted March, 010. The associate editor coordiatig the review of this paper ad approvig it for publicatio was D. Gesbert. This work was supported by NSF Grats CCF , CCF , CNS , ad CCF , ad preseted i part at the 4d Asilomar Coferece o Sigals, Systems ad Computers, Pacific Grove, CA, October 008 [1]. J. Yag is with the Departmet of Electrical ad Computer Egieerig, Uiversity of Marylad, College Park, MD, USA ( yagjig@umd.edu). S. Ulukus is with the Departmet of Electrical ad Computer Egieerig, Uiversity of Marylad, College Park, MD, USA ( ulukus@umd.edu). Digital Object Idetifier /TWC /10$5.00 c 010 IEEE Similar goals have bee udertake by various authors i recet years. Referece [] cosiders a time-slotted system with N queues ad oe server. The legth of the slot is equal to the trasmissio time of a packet i the queue. I each slot, the cotroller allocates the server to oe of the coected queues, such that the average delay i the system is miimized. The authors develop a algorithm amed logest coected queue (LCQ), where the server is allocated to the logest of all coected queues at ay give slot. The authors prove that i a symmetric system, LCQ algorithm miimizes the average delay. Referece [] does ot cosider the issue of power cosumptio durig trasmissios. Referece [3] combies iformatio theory ad queueig theory i a multi-access commuicatio over a additive Gaussia oise chael. Authors cosider a cotiuous time system, where the arrival of packets is a Poisso process, ad the service time required for each packet is radom. Oce a packet arrives, it is trasmitted immediately with a fixed power, i.e, there are o queues at the trasmitters. Each trasmitter-receiver pair treats the other active pairs as oise. Because of the iterferece from the other trasmitters, the service rate for each packet is a fuctio of the umber of active users i the system. Referece [3] derives a relatioship betwee the average delay ad a fixed probability of error requiremet. Refereces [4] [1] cosider the data trasmissio problem from both iformatio theory ad queueig theory perspectives. Referece [4] (see also [5], [6]) aims to miimize the average delay through rate allocatio i a multi-access sceario i additive Gaussia oise. Ulike [3], i the settig of [4], packets arrive radomly ito the buffers of the trasmitters. Whe the queue at a trasmitter is ot empty, it trasmits a packet with a fixed power. Simultaeously achievable rates are characterized by the capacity regio of a multiple-access chael, which, for the o-fadig Gaussia case, is a petago. The purpose of [4] is to fid a operatig poit o the capacity regio of the correspodig multiple-access chael such that the average delay is miimized. The author develops the loger-queue-higher-rate (LQHR) allocatio strategy i the symmetric multi-access case, which is show to miimize the average delay of the packets. The LQHR allocatio scheme selects a extreme poit (i.e., a corer poit) i the multiaccess capacity regio. Referece [7] (see also [8]) cosiders the problem of rate/power cotrol i a sigle-user commuicatio over a fadig chael. It cosiders a discrete-time model, ad ivestigates adaptig rate/power i each slot accordig to the queue legth, source state ad chael state. The objective

2 YANG ad ULUKUS: DELAY-MINIMAL TRANSMISSION FOR AVERAGE POWER CONSTRAINED MULTI-ACCESS COMMUNICATIONS 755 is to miimize the average power with delay costraits. It discusses two trasmissio models. I the first model, the trasmissio time of a codeword is fixed, while the rate varies from block to block. I the secod model, the trasmissio time for each codeword varies. It formulates the problem ito a dyamic programmig problem ad develops a delay-power tradeoff curve. Refereces [9] [1] have similar formulatios. Referece [9] uses dyamic programmig to umerically calculate the optimal power/rate cotrol policies that miimize the average delay i a sigle-user system uder a average power costrait. Referece [10] derives bouds o the average delay i a system with a sigle queue cocateated with a multi-layer ecoder. Referece [11] formulates the power costraied average delay miimizatio problem ito a Markov decisio problem ad aalyzes the structure of the optimal solutio for a sigle-user fadig chael. Referece [1] proposes a dyamic programmig formulatio to fid optimal power, chael codig ad source codig policies with a delay costrait. As i [7], i these papers as well, because of the large umber of possible rate/power choices at each stage, it is almost impossible to get aalytical optimal solutios. Referece [13] cosiders a cogitive multiple access system. I the model of [13], the primary user (PU) always trasmits a packet durig a slot wheever its queue is ot empty. The secodary user (SU) always trasmits whe the PU is idle, ad it trasmits with some probability (which is a fuctio of its ow queue legth) whe the PU is active. The receiver operates at the corer poit of the multiple access chael capacity regio where the SU is decoded first ad the PU is decoded ext, so that eve though the SU experieces iterferece from the PU, the PU is always decoded iterferece-free. Referece [13] aims to miimize the average delay through cotrollig the trasmissio probability of the SU. It formulates the problem as a oe-dimesioal Markov chai ad derives a aalytical result to miimize the average delay of the SU uder a average power costrait. I this paper, we geeralize [13] to a two-user multi-access system, where both users have equal priority. Our goal is to miimize the average delay of the packets i the system uder a average power costrait for each user. As i [7], [9], [11], [13], we cosider a discrete-time model. We divide the trasmissio time ito time slots. Packets arrivig at the trasmitters are stored i the queues at each trasmitter. I each slot, each user decides o a trasmissio rate based o the curret legths of both queues. Ulike [7], [9], [11], where the rate per slot is a cotiuous variable, we restrict the trasmissio rate for each user i a slot to be either zero or oe packet per slot. We defie the probabilities of choosig each trasmissio rate pair, which ca be (0, 0), (0, 1), (1, 0) or (1, 1), for each give pair of queue legths. Our objective is to fid a set of trasmissio probabilities that miimizes the average delay while satisfyig the average power costraits for both users. As i [13], there are two mai reasos that we itroduce trasmissio probabilities: First, a radomized policy is more geeral tha a determiistic policy; probability selectios of 0 ad 1 correspod to a determiistic policy, which is a special case of the radomized policy. Secodly, sice we caot choose arbitrary departure rates i each slot, the use of trasmissio probabilities eables us to cotrol the average rate per slot at a fier scale. Compared to [7], [9], [11], our model has a more restricted policy space at each stage, however, this model eables us to costruct a twodimesioal discrete-time Markov chai ad evetually gives us a closed-form optimal solutio. I the rest of this paper, we first express the average delay ad the average power cosumed for each user as fuctios of the trasmissio probabilities ad steady state distributio of the queue legths. We the trasform our problem ito a liear programmig problem, ad derive the optimal trasmissio scheme aalytically. We show that the optimal trasmissio policy has a threshold structure, i.e., if the sum of the queue legths exceeds a threshold, both users trasmit a packet from their queues, ad if the sum of the queue legths is smaller tha a threshold, oly oe user, which has the larger queue legth, trasmits a packet from its queue, while the other user remais silet (equal queue legth case is resolved by flip of a potetially biased coi). We provide a rigorous mathematical proof for the optimality of the solutio. We also provide umerical examples for both symmetric ad asymmetric settigs. II. SYSTEM MODEL A. Physical Layer Model We cosider a discrete-time additive Gaussia oise multiple-access system with two trasmitters ad oe receiver. The received sigal is Y = h 1 X 1 + h X + Z (1) where X i is the sigal of user i, h i is the chael gai of user i, adz is a Gaussia oise with zero-mea ad variace σ. Here, h 1 ad h are real costats, with h 1 = h i geeral. I this two-user system, sice the multiple-access capacity regio is give as [14] R 1 1 ( log 1+ h ) 1P 1 σ () R 1 ( log 1+ h ) P σ (3) R 1 + R 1 ( log 1+ h ) 1P 1 + h P σ (4) the regio of feasible received powers is give by [15] h 1 P 1 σ ( R1 1) (5) h P σ ( R 1) (6) h 1 P 1 + h P σ ( (R1+R) 1) (7) I each slot, the trasmitters adjust their trasmitted powers to achieve the desired trasmissio rates. We assume that for each user, the average trasmitted power over all of the slots must satisfy a costrait. We deote the average power costraits for the first ad secod user as P 1avg ad P avg, respectively. B. Medium Access Cotrol (MAC) Layer Model I the MAC layer, we assume that packets arrive at the trasmitters at a uiform size of B bits per packet. We partitio

3 756 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 9, NO. 9, SEPTEMBER 010 Fig. 1. a1 a System model. user 1 user receiver the time ito small slots such that we have at most oe packet arrive ad/or depart durig each slot. Let a 1 [] ad a [] deote the umber of packets arrivig at the first ad secod trasmitters, respectively, durig time slot ; see Figure 1. We assume that the packet arrivals are i.i.d. from slot to slot, ad the probabilities of arrivals are Pr{a i [] =1} = θ i (8) Pr{a i [] =0} =1 θ i (9) where θ i is the arrival rate for user i, i =1,. There is a buffer with capacity N at each trasmitter to store the packets, where N is a fiite positive iteger. Oce the buffer is ot empty, the trasmitters decide to trasmit oe packet i the slot with some probability based o the curret legths of both queues. Let d 1 [] ad d [] deote the umber of packets trasmitted i time slot. The queue legth i each buffer evolves accordig to q 1 [ +1]=(q 1 [] d 1 []) + + a 1 [] (10) q [ +1]=(q [] d []) + + a [] (11) where (x) + deotes max(0,x). The departure rate for each queue i each slot is either zero or oe packet per slot, ad the decisio whether it should be zero or oe packet per slot depeds o the curret queue legths. Whe both queues are empty, the departure rates for both queues should be zero packet per slot. I all other situatios where there is at least oe packet i at least oe of the queues, the departure rates for both queues should ot be zero packet per slot simultaeously. This is because, keepig both trasmitters idle does ot save ay power, but causes uecessary delay. Therefore, i these situatios, there are three possible departure rate pairs: (d 1,d ) = (1, 0), (0, 1) or (1, 1), i.e., oe packet is trasmitted from queue 1 ad o packet is trasmitted from queue ; o packet is trasmitted from queue 1 ad oe packet is trasmitted from queue ; or, oe packet is trasmitted from each queue. We eumerate them as d 1,d,d 3. Whe the first queue legth is i ad the secod queue legth is j, wedefie the probabilities of choosig each pair of these departure rates as gij 1, g ij, g3 ij, respectively. Note that gij 1 + g ij + g3 ij =1. We also ote that gij 1, g ij, g3 ij,fori =0, 1,...,N ad j =0, 1,...,N are the mai parameters we aim to choose optimally i this paper. The state space of the Markov chai cosists of all possible pairs of queue legths. We deote the state as q (q 1,q ). Whe both of the queues are empty, i.e., q[] = (0, 0), trasmitters have o packet to sed, ad from (10)-(11), q[ +1] = a[]. The correspodig trasitio probabilities i this case are: Pr{q[ +1]=(0, 0) q[] =(0, 0)} =(1 θ 1 )(1 θ ) Pr{q[ +1]=(1, 0) q[] =(0, 0)} = θ 1 (1 θ ) Pr{q[ +1]=(0, 1) q[] =(0, 0)} = θ (1 θ 1 ) Pr{q[ +1]=(1, 1) q[] =(0, 0)} = θ 1 θ (1) Whe oe of the queues is empty, there is oly oe possible departure rate pair, which is either (0, 1) or (1, 0), depedig o which queue is empty. Therefore, from our argumet above, the departure probabilities should ot be free parameters, but must be chose as gi0 1 = g 0j =1. The correspodig trasitio probabilities are: Pr{q[ +1]=(i 1, 0) q[] =(i, 0)} =(1 θ 1 )(1 θ ) Pr{q[ +1]=(i 1, 1) q[] =(i, 0)} = θ (1 θ 1 ) Pr{q[ +1]=(i, 0) q[] =(i, 0)} = θ 1 (1 θ ) Pr{q[ +1]=(i, 1) q[] =(i, 0)} = θ 1 θ (13) A similar argumet is valid whe the first queue is empty, i.e., q[] =(0,j). Trasitio probabilities i this case ca be writte similar to (13). Whe either of the queues is empty, i.e., for q[] =(i, j), where 1 i, j N, the trasitio probabilities are: Pr{(i 1,j 1) (i, j)} = gij(1 3 θ 1)(1 θ ) Pr{(i 1,j+1) (i, j)} = gijθ 1 (1 θ 1) Pr{(i +1,j 1) (i, j)} = gijθ 1(1 θ ) Pr{(i, j +1) (i, j)} = gijθ 1 1θ (14) Pr{(i +1,j) (i, j)} = gijθ 1θ Pr{(i 1,j) (i, j)} = gijθ 3 (1 θ 1)+gij(1 1 θ 1)(1 θ ) Pr{(i, j 1) (i, j)} = gijθ 3 1(1 θ )+gij(1 θ 1)(1 θ ) Pr{(i, j) (i, j)} = gijθ 1 1(1 θ )+gijθ (1 θ 1)+gijθ 3 1θ For example, the first equatio i (14) is obtaied by otig that, for the ext queue state to be (i 1,j 1), we eed to trasmit oe packet from each queue ad we should have o arrivals to either of the queues. The probability of this evet is gij 3, probability of trasmittig oe packet from each queue, multiplied by (1 θ 1 ), probability of havig o arrivals to queue 1, ad (1 θ ), probability of havig o arrivals to queue. I this paper, we assume that the average power costraits are large eough to prevet ay packet losses. I order to prevet overflows, we always choose to trasmit oe packet from a queue wheever its legth reaches N. Therefore, we have gin 1 = g Nj = g3 NN =1. The two-dimesioal Markov chai is show i Figure. I [16], it is prove that, for all irreducible, positive recurret discrete-time Markov chais with state space S, there exists a statioary distributio {π s,s S}, which is give by the uique solutio to π s p sr = π r, s S π s =1 (15) s S It is also stated that for a reducible Markov chai with a sigle closed positive recurret aperiodic class ad a oempty set T, where for ay i T, the probability of gettig absorbed i the closed class startig from state i is 1, ad the steady

4 YANG ad ULUKUS: DELAY-MINIMAL TRANSMISSION FOR AVERAGE POWER CONSTRAINED MULTI-ACCESS COMMUNICATIONS 757 0, 0 0, 1 0,j 1 0, j 0,j+1 0,N 1 0, N ece from the other trasmitter, the trasmitted power for the active user eeds to satisfy 1, 0 1, 1 i 1, 0 i 1, 1 i+1, 0 i+1, 1 i 1,j 1 i, 0 i, 1 i,j 1 N 1, 0 N, 0 Fig.. N 1, 1 N, 1 1,j 1 i+1,j 1 N 1,j 1 N,j 1 1, j i 1, j i, j i+1, j N 1,j N, j Two-dimesioal Markov chai. state distributio exists. I our problem, we first assume that the statioary distributio exists for the optimal solutio. Oce we determie the solutio, we verify that the correspodig Markov chai has a uique statioary distributio. Let us defie the steady state distributio of this Markov chai as π =[π 00,π 01,,π 0N,π 10,,π NN ]. The, the steady state distributio must satisfy 1,j+1 i 1,j+1 i,j+1 i+1,j+1 N 1,j+1 N,j+1 1, N 1 i 1,N 1 i,n 1 i+1,n 1 N 1,N 1 N,N 1 1, N i 1, N i, N i+1, N N 1,N N, N πp = π, π1 =1 (16) where P is the trasitio matrix defied by the trasitio probabilities (1)-(14). We ca express the average umber of packets i the system as π ij(i + j). Accordig to Little s law [16], for a fixed sample path i a queueig system, if the limits of average waitig time W ad average arrival rate λ exist as time goes to ifiity, the the limit of average queue legth L exists ad they are related as L = λw. For our problem, i a system without overflow, these limits exist ad the average delay D is equal to 1 D = π ij (i + j) (17) θ 1 + θ where θ 1 + θ is the average arrival rate for the system. III. PROBLEM FORMULATION The trasmissio rate for both trasmitters durig a slot is either oe packet per slot or zero packet per slot. Equivaletly, the trasmissio rate is either B/τ bits/chael use or 0 bits/chael use, where τ is the umber of chael uses i each slot. We assume that i each slot we ca use codewords with fiite block legth to get arbitrarily close to the boudary of feasible powers ad achieve a satisfactory level of reliability. Next, let us cosider the power cosumptios durig each slot. Whe oly oe user trasmits, sice there is o iterfer- h i P i σ ( R 1) α (18) where R = B/τ. I order to miimize the power, the trasmitted power for the active user should be α/h i, depedig o which user is trasmittig. Whe both users trasmit simultaeously, the received powers should additioally satisfy h 1 P 1 + h P σ ( 4R 1) β (19) The feasible trasmitted power regio is show i Figure 3. Let us deote the received power pair as (β 1,β ). I order to miimize the trasmit power, this pair should be o the domiat face of the feasible power regio, i.e., β 1 + β = β. The, the correspodig trasmit power pair is (β 1 /h 1,β /h ). Note that differet operatig poits eed differet sum of trasmit powers. Thus, for ay state (i, j) = (0, 0), the average power cosumed for the first queue is 1 h 1 (gij 1 α+g3 ij β 1), while the average power cosumed for the secod queue is 1 h (gij α+g3 ij β ).Our goal is to fid the trasmissio policy, i.e., the probabilities gij k, k =1,, 3, i =0, 1,...,N, j =0, 1,...,N alog with the operatig poit (β 1,β ), such that the average delay is miimized, subject to a average power costrait for each user. Therefore, our problem ca be expressed as: 1 mi π ij (i + j) (0) g,β 1,β θ 1 + θ s.t. 1 π ij (g h ijα 1 + gijβ 3 1 ) P 1avg (1) 1 1 π ij (g h ijα + gijβ 3 ) P avg () πp = π, π1 =1 (3) gij 1 + g ij + g3 ij =1, =0, 1,...,N (4) gij k 0, =0, 1,...,N, k =1,, 3 (5) We ote that the state trasitio matrix P is filled with variables i (1)-(14) which deped o gij k s. Also, through (3), π ij s deped o gij k s, as well. Ulike [13], we have a two-dimesioal Markov chai, ad it does ot admit closedform expressios for the steady state distributio π ij siterms of gij k s. Therefore, solvig the above optimizatio problem becomes rather difficult. Our methodology will be to trasform our optimizatio problem ito a liear programmig problem, ad exploit its special structure to obtai the globally optimal solutio aalytically. IV. ANALYSISOFTHEPROBLEM Note that gij 1 +g ij +g3 ij =1for ay (i, j) = (0, 0), therefore π ij = π ij (gij 1 + g ij + g3 ij ).Defie x 00 = π 00, x k ij = π ijgij k, k =1,, 3, i =0, 1,...,N, j =0, 1,...,N.Sicegij k is the coditioal probability of choosig policy k whe the system is i state (i, j), x k ij ca be iterpreted as the ucoditioal probability of stayig i state (i, j) ad choosig policy k. Our aim is to fid optimal gij k s. However, as we will see, our aalysis will be more tractable with variables x k ijs. Oce we

5 758 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 9, NO. 9, SEPTEMBER 010 P 1 Therefore, if the average power costraits P 1avg ad P avg satisfy the followig iequalities P 1avg h 1 + P avg h (θ 1 + θ θ 1 θ )α + θ 1 θ β (33) Fig. 3. β/h 1 β 1 /h 1 α/h 1 α/h β /h Feasible power regio. β/h fid optimal x k ij s, we ca obtai optimal gk ij s through gij k x k ij = 3 (6) k=1 xk ij Let us costruct a vector of all of our ukows x = [x 00,x 1 01,x 01,x3 01,...,x3 NN ]T. First, we cosider the average power cosumptio whe average power costraits for both users are large eough such that each user is able to trasmit a packet durig a slot wheever its queue is ot empty. I this sceario, the correspodig Markov chai has four o-trasiet states, (0,0), (0,1), (1,0), (1,1), ad the statioary distributio is π 01 = θ (1 θ 1 ), π 00 =(1 θ 1 )(1 θ ), π 10 = θ 1 (1 θ ), π 11 = θ 1 θ (7) The average power cosumptio for each queue is We ote that P 1csmp = 1 h 1 (π 10 α + π 11 β 1 ) = 1 h 1 (θ 1 (1 θ )α + θ 1 θ β 1 ) (8) P csmp = 1 h (π 01 α + π 11 β ) = 1 h (θ (1 θ 1 )α + θ 1 θ β ) (9) P 1csmp h 1 + P csmp h =(θ 1 + θ θ 1 θ )α + θ 1 θ β (30) From Figure 3, we ote that β 1,β α, therefore, each idividual term i (30) must additioally satisfy P 1csmp 1 h 1 θ 1 α (31) P csmp 1 h θ α (3) P P 1avg 1 h 1 θ 1 α (34) P avg 1 h θ α (35) the we ca always fid a operatig poit (β 1,β ) such that P 1csmp P 1avg ad P csmp P avg, ad we achieve the miimal possible delay i the system, which is oe slot. The available power i this case is so large that the solutio is trivial. If P 1avg h 1 + P avg h < (θ 1 + θ θ 1 θ )α + θ 1 θ β (36) ad P 1avg ad P avg are large eough to prevet ay overflows, both power costraits should be tight. Therefore, from (1)-(), we have two equality power costraits, 1 (x 1 h ijα + x 3 ijβ 1 )=P 1avg (37) 1 1 (x h ijα + x 3 ijβ )=P avg (38) Because the average arrival rate must be equal to the average departure rate whe there is o overflow, we also have (x 1 ij + x 3 ij) =θ 1 (39) (x ij + x3 ij )=θ (40) Solvig (37)-(40), we obtai (β α)(p 1avg h 1 θ 1 α) β 1 = α + P 1avg h 1 + P avg h (θ 1 + θ )α (β α)(p avg h θ α) β = α + P 1avg h 1 + P avg h (θ 1 + θ )α x 1 ij = θ 1 P 1avgh 1 + P avg h (θ 1 + θ )α β α x ij = θ P 1avgh 1 + P avg h (θ 1 + θ )α β α x 3 ij = P 1avgh 1 + P avg h (θ 1 + θ )α β α (41) (4) (43) (44) (45) By joitly cosiderig the ormalizatio equatio i (3), we also have x 00 =1 (θ 1 + θ )(β α) (P 1avg h 1 + P avg h ) (46) β α Thus, we trasform our optimizatio problem i (0)-(4) ito mi x s.t. ( 3 ) x k ij(i + j) k=1 (θ1+ θ)(β α) (P1avgh1 + Pavgh) x 00 =1 β α x 1 P1avgh1 + Pavgh (θ1 + θ)α ij = θ 1 β α (47) (48) (49)

6 YANG ad ULUKUS: DELAY-MINIMAL TRANSMISSION FOR AVERAGE POWER CONSTRAINED MULTI-ACCESS COMMUNICATIONS 759 x ij = θ x 3 ij = P1avgh1 + Pavgh (θ1 + θ)α β α P1avgh1 + Pavgh (θ1 + θ)α β α (50) (51) Qx = 0, x k ij 0, =0, 1,...,N, k=1,, 3 (5) 0, 0 0, 1 0, 0, 3 1, 0 1, 1 1, 1, 3 whichisitermsofx k ij s. Here, Q is a (N +1) (4(N + 1) 3) matrix defied by matrix P. We get the equatios i (5) from (3) by substitutig π ij gij k for xk ij. The optimizatio problem i (47)-(5) is a liear programmig problem. I additio, we observe that, i the objective fuctio, all of the x k ijs with the same sum of idices share the same weight i + j. If we look ito the two-dimesioal Markov chai, this correspods to the states o the diagoals ruig from the upper right corer to the lower left corer. This motivates us to group the x k ijs alog the diagoals of the two-dimesioal Markov chai i Figure ad defie their sum, for the th diagoal, as y = (x 1 i, i + x i, i) (53) t = i=0 x 3 i, i (54) i=0 The, y 0, t 0, ad the objective fuctio i (47) is equivalet to N (y + t ) (55) We also get N flow-i-flow-out equatios betwee the diagoal groups. For =0, 1, wehave x 00 (θ 1 + θ θ 1 θ )=(y 1 + t )(1 θ 1 )(1 θ ) (56) (x 00 + y 1 )θ 1 θ =(y +t 3 )(1 θ 1 )(1 θ )+t (1 θ 1 θ ) (57) ad for =, 3,...,N, wehave y θ 1 θ =(y +1 +t + )(1 θ 1 )(1 θ )+t +1 (1 θ 1 θ ) (58) y N 1 θ 1 θ = t N (1 θ 1 θ ) (59) Figure 4 shows the trasitios betwee diagoal groups for a system with N =3; we use differet colors to distiguish the trasitios caused by differet departure rate pairs. We multiply both sides of the th equatio i (56)-(59) with z ad sum with respect to to obtai x 00 (θ 1 + θ θ 1 θ + θ 1 θ z) + ( θ 1 θ (1 θ 1 )(1 θ )z 1) N y z ( (1 θ 1 θ )z 1 +(1 θ 1 )(1 θ )z ) N t z =0 (60) Takig the derivative of (60) with respect to z ad lettig, 0 3, 0, 1,, 3 3, 1 3, 3, 3 Fig. 4. The trasitios betwee diagoal groups whe N =3. z =1,wehave ( ( N N ) 1 t = (θ 1 + θ 1) y θ 1 θ ( N ) +(1 θ 1 )(1 θ ) y (61) + ( 1 θ 1 θ +(1 θ 1 )(1 θ ) ) ( ) ) N t +x 00 θ 1 θ From the defiitio of y ad t i (53) ad (54), we ote N N y = t = N N i=0(x 1 i, i + x i, i )= (x 1 ij + x ij ) (6) x 3 ij (63) x 3 i, i = i=0 From (6) ad (63), ad usig (49)-(51), we coclude that N y ad N t are costats that deped o system parameters α, β, θ 1, θ ad P 1avg, P avg. Usig (6) ad (49)-(50), for future referece, let us defie N y = x 1 ij + x ij (64) = θ 1 + θ (P 1avgh 1 + P avg h (θ 1 + θ )α) β α Ψ Usig the defiitio of y, t ad (61), the objective fuctio i (47) becomes ( N N ) 1 (y + t ) = y + C (65) θ 1 θ 1 where C is a costat, ad θ 1 θ is positive. Therefore, miimizig the origial objective fuctio i (47) is equivalet to miimizig N y. Sice from (64) the sum of y s is fixed, ad y s are positive, ituitively, the optimizatio

7 760 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 9, NO. 9, SEPTEMBER 010 problem requires us to assig larger values to y s with smaller idices, without coflictig with the trasitio equatio costraits. V. THE MODIFIED OPTIMIZATION PROBLEM AND A TWO-STEP SOLUTION I this sectio, we will prove the followig mai result of our paper: If the average power costraits P 1avg, P avg are large eough to prevet ay packet losses, the delayoptimal policy has a threshold structure. Whe the sum of the queue legths is larger tha the threshold, both users should trasmit; whe the sum of the queue legths is smaller tha the threshold, oly the user with the loger queue should trasmit; the equal queue legth case ca be resolved through flip of a potetially biased coi. We propose to solve our origial optimizatio problem i two steps. I the first step, we will cosider the optimizatio problem i terms of y sadt s, where the objective fuctio is N y, ad the costraits are (64), (48), (56)-(59), ad positivity costraits o y sadt s. The objective fuctio of this optimizatio problem is exactly the same as that of our origial optimizatio problem i (47)-(5), however, our costraits are more leiet tha those of (47)-(5). First, (64) is weaker tha (49)-(51), as it imposes a costrait o the sum while (49)-(51) impose costraits o idividual terms. Secodly, the trasitio equatios i (5) are betwee all of the states i the two-dimesioal Markov chai, while the trasitio equatios i (56)-(59) are oly betwee the diagoal groups i the Markov chai. Fially, we do ot explicitly impose the sum costrait o t o the ew problem. These imply that, the result we obtai i the first step, i priciple, may ot be feasible for the origial problem. Therefore, i the secod step we allocate y s ad t s we obtai from the first step to x k ijs i such a way that the remaiig idepedet trasitio equatios i (5) are satisfied. We ote that (39) ad (40) ca be derived from (5), therefore, oce (5) is satisfied, (39) ad (40) will be satisfied. Together with (64), we ca make sure that (49)-(51) are all satisfied. Therefore, if we ca fid a valid allocatio i the secod step, we will coclude that the solutio foud i the first step is a feasible solutio to our origial problem. Sice the problem we solve i the first step has the cost fuctio of our problem, but is subject to more leiet costraits, its solutio, i priciple, may be better tha the solutio of our origial problem. However, whe we prove that the solutio we obtai i our first step is withi the feasible set of our origial problem, we will have solved our origial problem. I additio, oce we prove the optimality of the solutio i the first step, it will be globally optimal for the origial problem. First, we will miimize N y subject to (64), (48), (56)-(59), ad y, t 0. This meas that we will allocate Ψ to y siawaytomiimize N y. This will require us to allocate larger values to y s with smaller, while makig sure that (64), (48), ad (56)-(59) are satisfied. We state the result of our first step i the followig theorem. Theorem 1 The optimal solutio of the problem mi N y s. t. (64), (48), (56) (59), ad y 0,t 0, (66) has a threshold structure. I particular, there exists a threshold such that for <, t =0ad for >, y =0. The proof of this theorem is give i Appedix A. I the followig, we cosider the trasitio equatios withi groups for each state. Sice addig more costraits caot improve the optimizatio result, if we ca fid a way to allocate y sadt stox k ijs, such that all of the remaiig trasitio equatios are satisfied, the we will coclude that the assigmets we obtaied i the first step are actually feasible for the origial problem. Therefore, ext, i our secod step, we focus o the assigmet of the y sadt s foud i the first step to x k ij s. First, we use a simple example to illustrate the procedure of allocatio withi each group, the, we geeralize the procedure to arbitrary cases. I this simple example, we assume that N =4. Assume that after the group allocatio, we obtaied y 1,...,y 5 ad t 5,t 6 =0, ad the rest of the y sadt s are equal to zero. I order to keep the allocatio simple, whe we assig y 3, y 5, t 5 i each group, we assig them oly to two states: (1, ), (, 1) ad (, 3), (3, ), respectively; while we assig y 4 to three states: (1, 3), (, ), (3, 1), ad we assig t 6 to a sigle state (3, 3). Figure 5 illustrates the allocatio patter withi groups. We do ot assig ay values to the states with dotted circles. The dotted states will be trasiet states after the allocatio. We eed to guaratee that the ozero-valued states oly trasit to other ozero-valued states. This requires us to set x 1 1 = x 1 = x 1 3 = x 3 =0, ad x 1 13 = x 3 13 = x 31 = x 3 31 = 0. The valid trasitios are represeted as arrows i Figure 5. We ca see that the trasitios are withi the positive recurret class. The, let us examie each group ad fid trasitio equatios to be satisfied for each state. For states (0, 1), (0, ), (1, ), (1, 3), (, 3), the trasitio equatios to be satisfied are x 01(1 θ (1 θ 1 )) =(x 00 + x 1 10)θ (1 θ 1 ) +(x 0 + x1 11 )(1 θ 1)(1 θ ) x 0 (1 θ (1 θ 1 )) =x 1 11 θ (1 θ 1 ) x 1(1 θ (1 θ 1 )) =(x 0 + x 1 11)θ 1 θ + x 1 1θ (1 θ 1 ) +(x 13 + x1 + x3 3 )(1 θ 1)(1 θ ) x 13(1 θ (1 θ 1 )) =(x x 3 3)θ (1 θ 1 ) x 3 (1 θ (1 θ 1 )) +x 3 3 (1 θ 1θ )=(x 13 + x1 )θ 1θ +(x x 3 33)θ (1 θ 1 ) (67) We have five more similar trasitio equatios for states (0, 1), (0, ), (1, ), (1, 3), (, 3). All the ukow variables are iteractig with each other through these equatios. How to fid a allocatio satisfyig all of these equatios simultaeously becomes rather difficult. After simple maipulatios,

8 YANG ad ULUKUS: DELAY-MINIMAL TRANSMISSION FOR AVERAGE POWER CONSTRAINED MULTI-ACCESS COMMUNICATIONS 761 equatios i (67) become equivalet to x 01 =(x 00 +x x 01)θ (1 θ 1 )+(x 0 + x 1 11)(1 θ 1 )(1 θ ) x 0 =(x x 0 )θ (1 θ 1 ) x 1 =(x 0 + x 1 11)θ 1 θ +(x 1 + x 1 1)θ (1 θ 1 ) +(x 13 + x1 + x3 3 )(1 θ 1)(1 θ ) x 13 =(x 1 + x 13 + x 3 3)θ (1 θ 1 ) x 3 =(x 13 + x1 )θ 1θ +(x x 3 + x3 33 )θ (1 θ 1 ) x 3 3(1 θ 1 θ ) (68) 0, 0 0, 1 0, 0, 3 0, 4 1, 0 1, 1 1, 1, 3 1, 4, 0, 1,, 3, 4 Observig the right had sides of (68), we ote that, x 00, x x 10, x 1 + x1 1, x1 3 + x 3, x3 33 are kow, therefore, the allocatio for states (0, 1), (0, ), (1, ), (1, 3), (, 3) depeds oly o the values of x 0 + x1 11, x1 + x 13,adx3 3. Similarly, the allocatio for states (1, 0), (, 0), (, 1), (3, 1), (3, ) also depeds o the values of x x 11, x + x 1 31, ad x 3 3 oly. Sice y =(x 0 + x1 11 )+(x1 0 + x 11 ) (69) y 4 =(x 1 + x 13)+(x + x 1 31) (70) t 5 = x x3 3 (71) the allocatio actually depeds o how we split y, y 4 ad t 5 betwee (x 0 + x1 11 ) ad (x1 0 + x 11 ), (x1 + x 13 ) ad (x + x1 31 ), x3 3 ad x3 3, respectively. Oce we fix thevalues of x 0 + x 1 11, x 1 + x 13, adx 3 3, we obtai the values of all of the states, completig the allocatio. We ote that there is more tha oe feasible allocatio withi groups, ad for each feasible allocatio, all of the trasitio equatios are satisfied, ad the power costraits are satisfied as well. I order to keep the solutio simple, we let Pluggig these ito (68), we get x 0 + x1 11 = y / (7) x 1 + x 13 = y 4 / (73) x 3 3 = t 5/ (74) x 01 =(x 00 + y 1 )θ (1 θ 1 )+ 1 y (1 θ 1 )(1 θ ) x 0 =1 y θ (1 θ 1 ) x 1 = 1 y θ 1 θ + y 3 θ (1 θ 1 )+ 1 (y 4 + t 5 )(1 θ 1 )(1 θ ) x 13 =1 (y 4 + t 5 )θ (1 θ 1 ) x 3 = 1 y 4θ 1 θ +(y 5 + t 6 )θ (1 θ 1 ) 1 t 5(1 θ 1 θ ) (75) Goig back to (7)-(73), we obtai x 1 11 = 1 y (1 θ (1 θ 1 )) x 1 = 1 y 4 1 (y 4 + t 5 )θ (1 θ 1 ) (76) Sice y t +1 ρ/δ, we ca easily verify that x 3 0, x1 0. The allocatio for the remaiig half of the states has a similar structure. Thus, each state has a positive value, ad the allocatio is feasible. Oce we obtai the values of x k ijs, we ca compute the 3, 0 3, 1 3, 4, 0 4, 1 4, 4, 3 4, 4 Fig. 5. Example: allocatio withi groups whe N =4. trasmissio probabilities usig gij k x = k ij 3. Here, we k=1 xk ij have g θ (1 θ 1 ) = (77) θ (1 θ 1 ) θ 1 (1 θ ) g11 1 θ 1 (1 θ ) = (78) θ (1 θ 1 ) θ 1 (1 θ ) g 1 y 4 (y 4 + t 5 )θ (1 θ 1 ) = (79) y 4 (y 4 + t 5 )(θ (1 θ 1 )+θ 1 (1 θ )) g y 4 (y 4 + t 5 )θ 1 (1 θ ) = (80) y 4 (y 4 + t 5 )(θ (1 θ 1 )+θ 1 (1 θ )) We observe that a threshold structure exists. I this example, the threshold is 5. Whe the sum of the two queue legths is greater tha 5, both users trasmit durig a slot. Whe the sum of the two queue legths is less tha 5, oly oe user with loger queue trasmits durig a slot; i this case, if both queue legths are the same, users trasmit accordig to probabilities i (77)-(80). Followig steps similar to those i the example above, we ca assig y sadt stox k ijs ad obtai a feasible allocatio for geeral settigs. The followig theorem states this fact formally. Theorem For the y s ad t s obtaied i the first step, there always exists a feasible x k ij assigmet, such that xk ij s are positive ad satisfy all of the trasitio equatios. The proof of this theorem is give i Appedix B. Sice this is a costructive proof, it also gives the exact method by which y sadt s are assiged to x k ij s. Therefore, i order to prove the optimality of the x k ij assigmet, it suffices to prove the optimality of the solutio of the first step. The followig theorem proves the optimality of the first step. 3, 3 3, 4

9 76 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 9, NO. 9, SEPTEMBER 010 Theorem 3 The allocatio scheme i Theorem 1 miimizes the average delay i the system. The proof of this theorem is give i Appedix C. I summary, the two-step allocatio scheme is feasible ad optimal for our origial problem. The trasitio probabilities ca be computed oce we determie the allocatio for each sate. From our allocatio, we ote that there exists a threshold, where is the largest group idex such that y =0.We have t > 0 oly whe. Sicegij k = x k ij 3,we k=1 xk ij have gij 3 =1whe >. Whe<, wehaveg1 ij =1if i>jad gij =1if i<j. The, for, ad is eve, we have g/,/ 1 = y (y + t +1 )θ (1 θ 1 ) y (y +t +1 )(θ (1 θ 1 )+θ 1 (1 θ ))+t (81) g/,/ = y (y + t +1 )θ 1 (1 θ ) y (y +t +1 )(θ (1 θ 1 )+θ 1 (1 θ ))+t (8) g/,/ 3 = y (y +t +1 )(θ (1 θ 1 )+θ 1 (1 θ ))+t (83) If t,t +1 =0, which happes whe < 1, (81)-(83) reduce to g/,/ 1 = 1 θ (1 θ 1 ) (84) θ (1 θ 1 ) θ 1 (1 θ ) g/,/ = 1 θ 1 (1 θ ) (85) θ (1 θ 1 ) θ 1 (1 θ ) Therefore, if the sum of the two queue legths is greater tha, both users should trasmit oe packet durig the slot. If the sum of the two queue legths is less tha, oly the user with the loger queue trasmits oe packet i the slot ad the other user remais silet; if i this case both queues have the same legth, the the probability that the first user trasmits oe packet while the secod user keeps silet 1 θ is (1 θ 1) θ (1 θ 1) θ 1(1 θ ), ad the probability that the secod user trasmits oe packet while the first user keeps silet 1 θ is 1(1 θ ) θ (1 θ 1) θ 1(1 θ ). Whe the system is symmetric, i.e., θ 1 = θ, these probabilities become 1/ ad 1/. VI. NUMERICAL EXAMPLES Here we give simple examples to show how our allocatio scheme works. We choose N =10, i.e., each queue has a buffer of size 10 packets. Therefore, the joit queue sates is represeted by a Markov chai. First, we cosider the symmetric sceario, where θ 1 = θ = θ, h 1 = h = h ad P 1avg = P avg = P avg. We assume the arrival rate θ =1/, ad the power levels α =1, β =3. Therefore, we have η =3, δ =1, ρ =3. From the aalysis, we kow that if P avg 5/8, the average delay is oe slot, which is the miimal possible delay i the system. If P avg = 9/16, we have x 00 = 1/8, x1 ij = x ij =3/8, x3 ij =1/8. Therefore, Ψ=3/4. Followig our allocatio scheme, we have y 1 =3/8, y =3/8, t 3 =1/8. The, we eed to allocate these withi groups. We start with y 1. Because of the symmetry of the settig, we simply let x 1 10 = x 01 = y 1 /=3/16, x 3 1 = x 3 1 = t t 3 /=1/16. The, we cosider y.wealsoletx 1 0 = x 0, x 1 11 = x 11. This symmetric allocatio guaratees that the flow equatios for states (0, 1) ad (1, 0) are satisfied. The values of x 1 0 ad x1 11 also deped o the allocatio of t 3.Thestate (, 0) must satisfy the trasitio equatio x 1 0 ( θ(1 θ)+θ +(1 θ) ) =(x 11 + x 3 1)θ(1 θ) Together with the symmetric allocatio, we have x x 11 = y /=3/16 Solvig these equatios, we get the allocatio for the secod group as x 1 0 = x 0 =1/16, x 11 = x 1 11 =1/8 We see that the two values are positive, thus feasible. The, the trasmissio probabilities are g11 1 = g 11 =1/, g3 1 = g1 3 =1. The threshold of the sum of the queue legths is i this case. If the sum of the queue legths is greater tha, both users trasmit, if the sum of the queue legths is less tha or equal to, oly the user with the loger queue trasmits ad the other user remais silet; if both queues have oe packet i their queues, each queue trasmits with probability 1/ while the other queue remais silet. If P avg = 17/3, we have x 00 = 1/16, x1 ij = x ij = 7/16, x3 ij = 1/16. Therefore, Ψ = 7/8. Followig our allocatio scheme, we have y 1 =3/16, y = y 3 =1/4, y 4 =3/16, t 5 =1/16. The, we assig these withi groups. For y 1,wesimplyletx 1 10 = x 01 = y 1/ =1/3. The, cosiderig to allocate y,wehavex 1 0 = x 0 =1/3, x 11 = x1 11 =3/3. After completig the allocatio, we have x 1 1 = x 1 =1/8, x 1 31 = x 13 =1/3, x = x 1 =1/16, x 3 3 = x 3 3 = 1/3. The trasmissio probabilities are g11 1 = g 11 = g1 = g =1/, g1 10 = g 01 = g1 0 = g 0 = g1 1 = g1 = g13 1 = g31 = g3 3 = g3 3 =1. The threshold of the sum of the queue legths is 4 i this case. If the sum of the queue legths is greater tha 4, both users trasmit, if the sum of the queue legths is less tha or equal to 4, oly the user with the loger queue trasmits ad the other user remais silet; if both queues have equal legth, which is either 1 or i this case, each queue trasmits with probability 1/ while the other queue remais silet. We compute the average delay as a fuctio of average power for θ =0.5, θ =0.48 ad θ =0.46, ad plot them i Figure 6. We observe that it is a piecewise liear fuctio, ad each liear segmet correspods to the same threshold value. This is because based o our optimal allocatio scheme, for a fixed threshold value, the objective fuctio is a liear fuctio i x 00, thus it is liear i P avg.ifp avg icreases, D avg decreases, ad the threshold decreases as well. The miimum value of P avg o each curve correspods to the maximum threshold, which is 19 i this example. This is also the miimum amout of average power required to prevet ay overflows. We also observe that the delay-power tradeoff curve is covex, which is cosistet with the result i [7]. We ote that although these three values of θ are close to each other, the average delay varies sigificatly. This is because the average delay is ot a liear fuctio of θ. For the asymmetric sceario, we assume θ 1 =1/, θ =

10 YANG ad ULUKUS: DELAY-MINIMAL TRANSMISSION FOR AVERAGE POWER CONSTRAINED MULTI-ACCESS COMMUNICATIONS θ=0.50 θ=0.48 θ=0.46 remais silet; if both queues have the same legth, oly oe of the queues trasmits with a probability which depeds o the arrival rates to both queues while the other queue remais silet. D avg Fig P avg The average delay versus average power i the symmetric sceario. 1/3, theη =, δ =1/, ρ =5/. We assume h 1 =1, h =. From (33), we kow that if P 1avg h 1 + P avg h 1, P 1avg 1/, P avg /3, the each user ca always trasmit a packet wheever its queue is ot empty, ad the average delay is oe slot. If P 1avg = 19/36, P avg = 13/18, the P 1avg h 1 + P avg h = 8/9. Pluggig these ito (41)-(48), we have β 1 = 1/, β = 1/, 1 x1 ij = 4/9, x1 ij = 5/18, 3 x1 ij =1/18, x 00 =/9. The, Ψ=13/18. Followig the group allocatio scheme, we have y 1 =4/9, y =5/18, t 3 =1/18. The, we eed to assig them withi groups. From (118)-(14), we get x 01 =1/6, x1 10 =5/18, x 0 =1/36, x 1 11 =4/36, x 11 =3/36, x 1 0 =/36,adx 3 1 = x 3 1 =1/18. The trasmissio probabilities are g11 1 = 4/7, g 11 = 3/7, g10 1 = g 01 = g1 0 = g 0 = g3 1 = g3 1 =1. The threshold is. If the sum of the queue legths is greater tha, both users trasmit, if the sum of the queue legths is less tha or equal to, oly the user with the loger queue trasmits ad the other user remais silet; if both queues have oe packet i their queues, the first queue trasmits with probability 4/7, ad the secod queue trasmits with probability 3/7. VII. CONCLUSIONS We ivestigated the average delay miimizatio problem for a two-user multiple-access system with average power costraits for the geeral asymmetric sceario, where users have arbitrary powers, chael gais, ad arrival rates. We cosidered a discrete-time model. I each slot, the arrivals at each queue follow a Beroulli distributio, ad we trasmit at most oe packet from each queue with some probability. Our objective is to fid the optimal set of departure probabilities. We modeled the problem as a two-dimesioal Markov chai, ad miimized the average delay through cotrollig the departure probabilities i each time slot. We trasformed the problem ito a liear programmig problem ad foud the optimal solutio aalytically. The optimal policy has a threshold structure. Wheever the sum of the queue legths exceeds a threshold, both queues trasmit oe packet durig the slot, otherwise, oly oe of the queues, which is loger, trasmits oe packet durig the slot ad the other queue APPENDIX A. The Proof of Theorem 1 Let us defie η = θ 1 + θ θ 1 θ (1 θ 1 )(1 θ ) θ 1 θ δ = (1 θ 1 )(1 θ ) 1 θ 1 θ ρ = (1 θ 1 )(1 θ ) The, (56)-(59) are equivalet to (86) (87) (88) x 00 η = y 1 + t (89) (x 00 + y 1 )δ =(y + t 3 )+t ρ (90) ad for =, 3,...,N, y δ =(y +1 + t + )+t +1 ρ (91) y N 1 δ = t N ρ (9) The optimizatio requires us to assig larger values to y s with smaller idices as much as possible. Examiig (89)- (9), we ote that for fixed x 00, maximizig y 1, y,... requires us to set t, t 3,... to zero. Therefore, we choose y 1 = x 00 η (93) y =(x 00 + y 1 )δ (94) y = y 1 δ, t =0, =1,,..., (95) where is the largest iteger satisfyig y < Ψ. Let Δ=Ψ y. We eed to check that all of the group trasitio equatios are satisfied. Assume that >. IfΔ=y δρ/(δ + ρ), thelet y +1 =Δ, ad y =0, = +,...,N 1 (96) t + = y +1δ/ρ, ad t =0, = + (97) We ca verify that after this allocatio, group trasitio equatios (56)-(59) are satisfied. We also ote that Ψ is allocated to {y } +1, amog which, {y } attai their maximum possible values. Therefore, the objective fuctio achieves its miimal possible value for the first step. If Δ = y δρ/(δ + ρ), if we assig it to y +1 directly, the group trasitio equatios are ot satisfied automatically. I order to satisfy the group trasitio equatios, we eed to do some adjustmets. If Δ >y δρ/(δ + ρ), we assig Δ to y +1 ad y + proportioally. Specifically, we let y +1 = Δ(ρ + δ)+y δρ ρ y + = Δ(ρ + δ)ρ y δρ ρ t + = y δ(δρ + δ + ρ) Δ(ρ + δ) ρ (98) (99) (100)

11 764 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 9, NO. 9, SEPTEMBER 010 t +3 = Δ(ρ + δ)δ y δ ρ ρ (101) Sice y δ > Δ > y δρ/(δ + ρ), we ca verify that each value above is positive, ad the sum costrait ad the group trasitio equatios are satisfied. Amog the o-zero {y } +, although {y } attai their maximum, y +1 does ot. Therefore, differet from the first sceario, i this case, we caot immediately claim that the result is optimal. We will give the mathematical proof for the optimality of this assigmet later. If Δ <y δρ/(δ +ρ), we eed to remove some value from y ad assig it to y +1 to satisfy the equatios. Defie Δ =Δ+y ad assig Δ to y ad y +1 as follows y = Δ (ρ + δ)+y 1δρ ρ (10) y +1 = Δ (ρ + δ)ρ y 1δρ ρ (103) t +1 = y 1δ(δρ + δ + ρ) Δ (ρ + δ) ρ (104) t + = Δ (ρ + δ)δ y 1δ ρ ρ (105) Sice y 1δ <Δ <y 1δ(δρ/(δ + ρ)+1), we ca also verify that each value above is positive, ad the sum costrait ad the group trasitio equatios are satisfied. Similar to the secod case, we caot immediately claim that this result is optimal because after the adjustmet, y does ot achieve its maximum value. We will give the proof of optimality later. Whe =1, the allocatio will be i a differet form. If Δ (x 00 + y 1 )δρ/(δ + ρ), the we eed to use (x 00 + y 1 ) istead of y i (96)-(101). If Δ < (x 00 + y 1 )δρ/(δ + ρ), the y = Ψ(ρ + δ)+x 00(η δ)ρ ρ (106) y +1 = Ψ(ρ + δ)ρ x 00(η δ)ρ ρ (107) t +1 = x 00(ηδρ + ηδ + ηρ + δρ) Ψ(ρ + δ) ρ (108) t + = Ψ(ρ + δ)δ x 00(η δ)δ ρ (109) Whe =,ifδ y δρ/(δ + ρ), the allocatio of Ψ has the same form as i (96)-(101). If Δ <y δρ/(δ + ρ), the we eed to use (x 00 +y 1 ) istead of y 1 i (10)-(105). B. The Proof of Theorem While we geeralize the simple example to a arbitrary settig, we follow the same basic allocatio patter. If is odd, we assig y ad t oly to two states ( +1, ) 1 ad ( 1, ) ( +1 ;if is eve, we assig y to three states: +1, 1), (, ) (, 1, +1), ad we assig t to a sigle state (, ). We illustrate the allocatio patter i Figure 7. We eed to make sure that the trasitios oly happe withi the positive recurret class. Therefore, whe is odd, we let x 1 1 = x, =0;whe is eve,, 1 we let x 1 1, +1 = x +1, 1 =0. The, let us examie the = N- N-1 Fig. 7. Allocatio patter withi groups. trasitio equatios for the states. For =1, we have x 01(1 θ (1 θ 1 )) = (x 00 + x x 3 11)θ (1 θ 1 ) +(x 0 + x x3 1 )(1 θ 1)(1 θ ) (110) For =, 3,...,if is eve, the trasitios betwee states are illustrated i Figure 8. The trasitio equatio for state ( 1, +1) is x 1, +1 (1 θ (1 θ 1 )) =(x 1, +x3, +1 )θ (1 θ 1 ) (111) If is odd, the trasitios betwee states are illustrated i Figure 9. The trasitio equatio for state ( 1, ) +1 is x 1, +1 =(x 3, +1 +(x 1 +1, +1 (1 θ (1 θ 1 )) + x x 1 1, 1 + x 1, +3 )θ 1 θ, +1 + x 3 +1, +3 (1 θ 1 θ ) N N+1 N+ * N-1 N )(1 θ 1 )(1 θ ) +(x 1 +1, + x , )θ +1 (1 θ 1 ) (11) After a trasformatio, (110) is equivalet to x 01 =(x 00 + x x 01 + x 3 11)θ (1 θ 1 ) +(x 0 + x x3 1 )(1 θ 1)(1 θ ) (113) where x 00 is kow, x x 01 = y 1, x 3 11 = t. For =, 3,...,whe is eve, (111) is equivalet to x 1, +1 =(x1, + x 1, +1 + x3, +1)θ (1 θ 1 ) (114) ad whe is odd, (11) is equivalet to x 1, +1 =(x 3, +1 + x 1 1 +(x 1 +1, +1 + x 1, +3 +(x 1 +1, 1 + x 1, +1, 1 + x x 3 +1 )θ 1θ x 3 1, +3, +1, +1 )(1 θ 1)(1 θ ) (1 θ 1θ ) )θ (1 θ 1) (115)

12 YANG ad ULUKUS: DELAY-MINIMAL TRANSMISSION FOR AVERAGE POWER CONSTRAINED MULTI-ACCESS COMMUNICATIONS 765 3, +1-1, -1, +1 1, 1 1, +1 1, +3, -1,, +1 +1, 3 +1, 1 +1, +1 +1, +3 +1, -1 +1, +3, 1 +3, +1 Fig. 8. The trasitios betwee states whe is eve. Fig. 9. The trasitios betwee states whe is odd. where x x, 1 1 = y, +1, x 3 +1 = t, The trasitio equatios for the remaiig half of the recurret states ca be expressed i a similar form. Therefore, the values of x k ij s are determied oly by the allocatio of y betwee x 1 +1, 1 + x, ad x 1, +1 + x1, is eve, ad the allocatio of t to x 3 +1, 1 whe is odd. If we let whe ad x 3 1, +1 x 1, + x 1, +1 = y /, whe is eve (116) x 3 1, +1 = t /, whe is odd (117) ad solve equatios (113)-(115), the, for =1, we obtai x 01 =(x 00 + y 1 + t )θ (1 θ 1 ) + 1 (y + t 3 )(1 θ 1 )(1 θ ) x 1 10 =(x 00 + y 1 + t )θ 1 (1 θ ) For =, 3,...,if is eve, we get + 1 (y + t 3 )(1 θ 1 )(1 θ ) (118) x 1, +1 = 1 (y + t +1 )θ (1 θ 1 ) (119) x 1 +1, 1 = 1 (y + t +1 )θ 1 (1 θ ) (10) x 1, = 1 y 1 (y + t +1 )θ (1 θ 1 ) (11) x, = 1 y 1 (y + t +1 )θ 1 (1 θ ) (1) ad if is odd, we have x 1, = 1 +1 y 1θ 1 θ +(y + t +1 )θ (1 θ 1 ) + 1 (y +1 + t + )(1 θ 1 )(1 θ ) 1 t (1 θ 1 θ ) (13) x 1 +1, = 1 1 y 1θ 1 θ +(y + t +1 )θ 1 (1 θ ) + 1 (y +1 + t + )(1 θ 1 )(1 θ ) 1 t (1 θ 1 θ ) (14) This completes the allocatio. Note that t =0oly whe is equal to +1, +, ad/or +3, depedig o the value of Δ. Whet +1 =0, it automatically disappears from the right had sides of (118)-(14). From the group trasitio equatios, we have y t +1 ρ /δ, ad it is easy to verify that all states have oegative assigmets ad the trasitio equatios are also satisfied i this case. Therefore, there always exists a feasible allocatio to satisfy all of the trasitio equatios with y sdefied through this allocatio scheme. C. The Proof of Theorem 3 I a covex optimizatio problem, where the iequality costraits are covex ad the equality costraits are affie, if x is such that there exists a set of Lagrage multipliers which together with x satisfy the KKT coditios, the x is a global miimizer for the problem [17] [18]. I the first step, we have a liear objective fuctio ad liear costraits. Therefore, if we prove that the poit achieved by the assigmet satisfies the KKT coditios, the we ca say that it is the global miimizer for our problem. I the allocatio scheme, if Δ = y δρ/(δ + ρ), the it is easy to prove that the resultig allocatio is optimal, sice every y, < achieves its maximum possible value. However, this is ot the case whe Δ = y δρ/(δ + ρ), because the secod to last ozero y does ot achieve its maximum. I the followig, we prove that our allocatio is optimal for this case as well. Defie y = [y 1,y,...,y N 1,t,...,t N 1,t N ]. The, the liear equality costraits, icludig the N group trasitio equatios ad the sum costrait ca be writte as a (N+1) (N 1) matrix form as follows x 00η δ ρ x 00δ 0 δ ρ y T = δ ρ Ψ which we write equivaletly as, Ay T = b (15)

13 766 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 9, NO. 9, SEPTEMBER 010 b T = ( x 00 η x 00 δ(1 + η) x 00 δ (1 + η) x 00 δ N 1 (1 + η) Ψ ) T ρ + δ (ρ + δ)δ ρ+ δ 1 0 A = (ρ + δ)δ N 3 (ρ + δ)δ N 4 (ρ + δ)δ N (ρ + δ)δ N (ρ + δ)δ N 3 (ρ + δ)δ N 4 ρ + δ (16) (17) by defiig b, A i (16) ad (17) at the top of the page. The Lagragia is expressed as L(y, λ, μ) =c T y λ T (Ay b) μ T y (18) where c =[1,,, N 1, 0, 0,, 0], λ R N+1 ad μ R 4N. We eed to prove that there exists a set of λ, μ associated with our allocatio y, such that they satisfy μ 0, μ T y =0 (19) y 0, Ay T = b (130) c = A T λ + μ (131) Cosider the y we obtaied with the algorithm. Let us cosider the case whe Δ <y δρ/(δ+ρ) first. The allocatio idicates that y > 0 oly whe =1,,..., +1,ad t > 0 oly whe = +1, +. Because of the complemetary slackess i (19), we obtai μ =0, =1,,..., +1, +N 1, +N (13) Pluggig this ito (131), ad solvig the equatios, we have λ = 1 ρ , =1,,..., +1 λ N+1 = ρ ρ +1 + ( μ +N = λ 1 +(ρ + δ) 1 i= ) λ i δ i + ρδ λ, =, 3,..., (133) Thus, we have λ < 0 whe, which guaratees the positiveess of {μ } +N =N.Wealsohave N i= + λ i δ i 1 = (ρ + δ)(ρ +1) (134) ad μ = 1 ρ λ, = +,...,N 1 ( ) N μ = λ 1 +(ρ + δ) λ i δ i, i= = +N +1,...,4N (135) We ca always fid a set of egative {λ i } N i= + to satisfy (134). Sice they are all egative, this guaratees that {μ } N 1 = + ad {μ } 4N = +N+1 are positive. Therefore, at the poit y, we ca always fid a set of multipliers satisfyig all of the KKT costraits. This proves that the allocatio our algorithm gives is a global miimizer. REFERENCES [1] J. Yag ad S. Ulukus, Delay-miimal trasmissio for average power costraied multi-access commuicatios, Asilomar Coferece, 008. [] L. Tassiulas ad A. Ephremides, Dyamic server allocatio to parallel queues with radomly varyig coectivity, IEEE Tras. If. Theory, vol. 39, pp , Mar [3] I. E. Telatar ad R. G. Gallager, Combiig queueig theory with iformatio theory for multiaccess, IEEE J. Sel. Areas Commu., vol. 13, pp , Aug [4] E. Yeh, Delay-optimal rate allocatio i multiaccess commuicatios: a cross-layer view, i Proc. IEEE Workshop o Multimedia Sigal Processig 00, pp , Dec. 00. [5] E. Yeh ad A. Cohe, Throughput optimal power ad rate cotrol for queued multi-access ad broadcast commuicatios, i Proc. IEEE Iteratioal Symposium o Iformatio Theory, p. 11, Jue 004. [6] E. Yeh, Miimum delay multi-access commuicatio for geeral packet legth distributios, i Proc. Allerto Coferece o Commuicatio, Cotrol, ad Computig, pp , Sep [7] R. A. Berry ad R. G. Gallager, Commuicatio over fadig chaels with delay costraits, IEEE Tras. If. Theory, vol. 48, pp , May 00. [8] R. Berry, Power ad delay trade-offs i fadig chaels, Ph.D. dissertatio, MIT, Jue 000. [9] I. Bettesh ad S. Shamai, Optimal power ad rate cotrol for miimal average delay: the sigle-user case, IEEE Tras. If. Theory, vol. 5, pp , Sep [10] A. Steier ad S. Shamai, O queueig ad multi-layer codig, Coferece o Iformatio Scieces ad Systems, Mar [11] M. Goyal, A. Kumar, ad V. Sharma, Power costraied ad delay optimal policies for schedulig trasmissio over a fadig chael, i Proc. IEEE Ifocom, vol. 1, pp , Mar [1] T. Holliday ad A. Goldsmith, Optimal power cotrol ad sourcechael codig for delay costraied traffic over wireless chaels, i Proc. IEEE It. Cof. Commuicatios, vol., pp , Apr. 00. [13] W. Che, K. B. Letaief, ad Z. Cao, A joit codig ad schedulig method for delay optimal cogitive multiple access, i Proc. IEEE ICC, 008. [14] T. M. Cover ad J. A. Thomas, Elemets of Iformatio Theory. New York: Joh Wiley ad Sos, Ic., [15] E. Uysal-Biyikoglu ad A. El Gamal, O adaptive trasmissio for eergy efficiecy i wireless data etworks, IEEE Tras. If. Theory, vol. 50, pp , Dec [16] V. G. Kulkari, Modelig ad Aalysis of Stochastic Systems. Chapma ad Hall/CRC, [17] A. Tits, Optimal Cotrol, lecture otes, Uiversity of Marylad, 00. [18] S. Boyd ad L. Vadeberghe, Covex Optimizatio. Uited Kigdom: Cambridge Uiversity Press, 004.

14 YANG ad ULUKUS: DELAY-MINIMAL TRANSMISSION FOR AVERAGE POWER CONSTRAINED MULTI-ACCESS COMMUNICATIONS 767 etworks. Jig Yag received the B.S. degree i electroic egieerig ad iformatio sciece from Uiversity of Sciece ad Techology of Chia, Hefei, Chia i 004. Sice the, she has bee a PhD graduate studet i the departmet of electrical ad computer egieerig at the Uiversity of Marylad, College Park. Her research iterests are i wireless commuicatio theory ad etworkig, multi-user iformatio theory, queueig theory ad optimizatio i wireless etworks, with particular focus o delay optimizatio i wireless commuicatios ad Seur Ulukus received the B.S. ad M.S. degrees i electrical ad electroics egieerig from Bilket Uiversity, Akara, Turkey, i 1991 ad 1993, respectively, ad the Ph.D. degree i electrical ad computer egieerig from Rutgers Uiversity, NJ, i Durig her Ph.D. studies, she was with the Wireless Iformatio Network Laboratory (WINLAB), Rutgers Uiversity. From 1998 to 001, she was a Seior Techical Staff Member at AT&T Labs-Research i NJ. I 001, she joied the Uiversity of Marylad at College Park, where she is curretly a Associate Professor i the Departmet of Electrical ad Computer Egieerig, with a joit appoitmet at the Istitute for Systems Research (ISR). Her research iterests are i wireless commuicatio theory ad etworkig, etwork iformatio theory for wireless etworks, sigal processig for wireless commuicatios ad security for multi-user wireless commuicatios. Seur Ulukus is a recipiet of the 005 NSF CAREER Award, ad a co-recipiet of the 003 IEEE Marcoi Prize Paper Award i Wireless Commuicatios. She serves/served as a Associate Editor for the IEEE TRANSACTIONS ON INFORMATION THEORY sice 007, as a Associate Editor for the IEEE TRANSACTIONS ON COMMUNICATIONS betwee , as a Guest Editor for the IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONSi , as the co-chair of the Commuicatio Theory Symposium at the 007 IEEE Global Telecommuicatios Coferece, as the co-chair of the Medium Access Cotrol (MAC) Track at the 008 IEEE Wireless Commuicatios ad Networkig Coferece, as the co-chair of the Wireless Commuicatios Symposium at the 010 IEEE Iteratioal Coferece o Commuicatios, as the co-chair of the 011 Commuicatio Theory Workshop, ad as the Secretary of the IEEE Commuicatio Theory Techical Committee (CTTC) i