Stability and Throughput of Buffered Aloha with Backoff

LEE AND DAI: STABILITY AND THROUGHPUT OF BUFFERED ALOHA WITH BACKOFF 1 S tability and Throughput of Buf fered Aloha with Backof f Tony T. Lee, Fellow , IEEE , and Lin Dai, Member , IEEE Abstract —This paper studies th e buffered Aloha with K -exponential back off collision resolution algorithms. The buffered Aloha network is modeled as a multi-queue single-se rver system. We adopt a widely used approach in packet switching systems to decompose the multi-queue sy stem into independent first-in-first-out (FIFO) queues, which are hinged together by the probability of success of h ead-of-line (HOL) packets. A unified method is devised to tackle th e stability and throughput problems of K -exponential backoff with any cutoff phase K . For networks with a finite number of node s, we show that the K -exponential backoff is stable if the retransmission factor is properly chosen from the stable region. The maximum stable throughput is derive d and demonstrated via examples of geometric retransmission ( K =1) and exponential backoff ( K = ∞ ). For networks with an infinite number of nodes, we show that geometric retrans mission is unstable, and the stable netw ork throughput of exponential backoff can only be achieve d at the cost of potential unbounded delay in each input queue. Furth ermore, we address the stability issue of the systems at the undesired stable point. All analyt ical results presen ted in this paper are verified an d confirmed by simulations. Index Terms —Random access, slotted Aloha, ex ponential backoff, geometric ret ransmission, stability I. I NTRODUCTION fundamental problem of m ulti-access communications is how to efficiently share t he channel resource among multiple users. From the Aloha network to today’s IEEE 802.11 Wi-Fi network, random access has proven to be a simple yet elegant solution; tran smit if there is a request, and back off if a collision occurs. The minimum coordination and distributed control have enabled random access to become one of the most widely deployed net work technologies used today [1-3]. Throughput analysis on random access networks can be traced back to Abramson’s landm ark paper [1]. Assuming an infinite num ber of nodes, Abramson proposed to model the aggregate traffic as a Poisson random variable with param eter G , which captures the essence of contentions am ong users without intricate analysis. The stead y-state equilibrium throughput G exp(- G ) derived from t his simplified m odel sheds useful Manuscript received April 22, 2008. Tony T. Lee is with the Departm ent of Information Engineering, Chinese University of Hong Kong, N.T., Hong Kong (e-mail: ttlee@ie.cuhk.edu.hk ). Lin Dai is with the Department of Elect ronic Engineering, City University of Hong Kong, Kowloon, Hong Kong (e-mail: lindai@cityu.edu.hk ). insight into m any aspects of network performance, such as the maxim um network throughput of e -1 when G =1. Research interests have bifur cated since then. On one hand, intense activities have fo cused on the stability analysis of Abramson’s Al oha protocol, under the original assum ption that the network is saturated and each node always has a packet to transmit, or an infi nite number of bufferless nodes. Most of these studies ignored queuing aspects, for i nstance, the offered load of each node, but emphasized the throughput of the whole network. Specifically, early work showed that slotted Aloha is unstable with an infinite population [4-5 ]. To stabilize a finite-node Aloha system, a dri ft approach was developed to analyze the system transition so that the retransmission probability co uld be adjusted accordingly. This requires the knowledge of the number of backlogged nodes [6-9]. Rather than investigati ng how to estimat e the backlogs [10-11], binary exponentia l backoff (BEB) algorithms were pr oposed to achieve stability by reducing the retransmission proba bility according to the num ber of collisions the p acket has experienced [12]. It was pro ven in [13] that BEB i s also unstable with an infinite num ber of nodes. Later, under a finite-node m odel, it was shown that BEB can be stable if the aggregate arriv al rate is sufficiently small [14]. Different upper bounds of the aggreg ate arrival rate have been developed since then [15-16] but none of them has become the consensus [17]. The stability i ssue of random access protocols remains an open problem . In the meantime, a great deal of effort has been made to establish a buffered Aloha model, which was initiated in [18] and further developed in [19-22]. With the interactions am ong the different queues taken i nto consideration, an n -node buffered Aloha system was m odeled as an n -dimensional random walk and the exact rate region for the two-node case was derived in [18]. Unfortunately, the general ization of this approach to an arbitrary n -node system encount ered tremendous difficulties. When n exceeds three, the in teractions among the queues become unm anageable and the problem is intractable even for the si mplest version of slot ted Aloha [23], let alone an Aloha system with exponential backoff. In this paper, the slotted Aloha is modeled as a multi-queue single-server system; each node is equipped with an infinite buffer and treated separately . A wi dely used approach in packet switching system s [24-26] is adopted to decompose the multi-queue system into indep endent FIFO queues with Bernoulli arrivals of rate λ packets per time slot. These A LEE AND DAI: STABILITY AND THROUGHPUT OF BUFFERED ALOHA WITH BACKOFF 2 decomposed Geo/G/1 queues are t hen hinged together by the service time that is determined by the probability of success p of HOL packets. The stability of networ ks with n nodes can be defined in terms of either the network throughput or dela y. However, the two definitions are not exactly t he same. Under the throughput definition, a network is stable if the network throughput ˆ out λ equals the aggregate input rate ˆ n λ λ = ; or the departure rat e equals the input rate. Under the delay definition, a network is stable if the offered load ρ of each input Geo/G/1 queue is strictly less than 1. It is obvious that the delay stability implies the throughput stability, but the reve rse is not necessarily true. A network with stable throughput that fails to ensure the delay bound is called pseudo-stabl e . The probability of success of a buffered Aloha with K -exponential backoff has one desired stable point at p L , one unstable equilibriu m point at p S and one undesired stable point at p A . Both p L and p S are roots of the characteristic equation of probability of success p, which is completely determined by the aggregate input rate ˆ λ . The undesired stable point p A , however, is dependent on the backoff parameters such as the retransmission factor q and the cutoff phase K . Our analysis is concentrated on the specifications of the stable regions of retransmissi on factor q . In the ab solute stable region, the network is guaranteed to converge to the desired stable point p L that ensures both throughput and delay stabilities. In a region with weaker assurance, called asymptot ic stable region , the stability of exponential backoff can be ensured with a very high probab ility. Outside these desired stable regi ons, we show that the network with exponential backoff may poise on t he undesired stable point p A with stable t hroughput but unbounded delay. A pseudo-stable regi on is introduced to mani fest the resilience of exponential backo ff in the face of transient fluctuations of input traffic. The stabl e regions and the associated maxi mum throughput in these regions of K -exponential backoff are outlined as follows. In particular, geometric retransmission ( K =1) and exponential backoff ( K = ∞ ) are two stereotypes of interest. Geometric Retransmission: For networks with a finite number of nodes n , the absolut e stable region is given by: ˆ ln (1 ) ˆ () S L L p p q n pn λ λ − ≤≤ − − and the maxim um stable throughput is e -1 . The network is unstable if the num ber of nodes n is infin ite, which agrees with previous studies [4-5] that t he slotted Aloha is inherently unstable as n →∞ . Exponential backoff: For networks with a finite num ber of nodes n , the absolute stable regi on is given by: ln 1 ˆ 1/ S L p p q n n λ − ≤≤ − − and the maxim um stable throughput is ln n / n. It has been shown in [14-16] that the network with binary exponential backoff (BEB) is stable if the arrival rate is sufficien tly small. Our result specifically shows t hat if the aggregate input rate ˆ λ is lower than /2 1 2 n ne − , then the retransmission factor q =1/2 is inside the absolute stable region. With an infi nite number of nodes n , the absolute stable region becomes empty . Furthermore, we show that in the following asym ptotic stable region ln 1 1 ˆ 1/ S L L L p p qp n n λ p − ≤≤ − − − , the maxi mum throughput e -1 is achievable with p robability 1- ε , and ε→ 0 as the num ber of nodes n → ∞ . If the number of nodes n is infinite, the asymptotic stable region shrinks to a single point q =1- p L , which coincides with th e result claimed in [17]. However, the mean del ay at this point would be unbounded and the network is pseudo-stable. Below the un stable equilib rium point p S , the probab ility of success will drift to the undesired stable point p A. . We prove that this undesired stable point p A converges to 1- q a s n →∞ , and t he pseudo-stable region is gi ven by: 11 L S pq p − ≤≤ − . In this regio n, the networ k is stable in terms of throughput, ˆ out ˆ λ λ = , while at the cost of unbounded m ean delay. Because of this pseudo-stability, exponential backoff is more adaptive to tran sient traffic fluctuati ons when compared to geometric ret ransmission. The network is unstable outside this pseudo-stable region as the throughput ˆ out λ < ˆ λ . The remainder of thi s paper is organized as follows. Section II establishes the queuing model and presents the preliminary analysis of buffered Aloha with backoff scheduling. Section III describes the absolute stab le region, inside wh ich the system can be guaranteed to converge to the desired stabl e point. Section IV introduces the asympto tic stability of exponential backoff. The analysis on the undesired stable point and the pseudo-stability of exponential are presented in Section V. The sim ulation results are provided in Sect ion VI and conclusions are summa rized in Section VII. The main notations used in this pap er are listed as follows: n : number of nodes K : cutoff phase q : retransmission factor (0< q <1) λ : input rate per node ˆ λ : aggregate input rate, also the steady -state network throughput. ˆ λ = n λ ρ : offered load of each node’s queue p : probability of success G : attempt rate p L : desired stable point . 0 ˆ exp{ ( )} L pW λ =− p S : unstable equilibriu m. 1 ˆ exp{ ( )} S pW λ − =− p A : undesired stable point q l : lower bound of retransmi ssion factor q q u : upper bound of retransmissi on factor q S : stable region of retransm ission factor q . S =[ q l , q u ] max_ ˆ S λ : maxi mum stable t hroughput. max_ ˆˆ sup SS λ λ = LEE AND DAI: STABILITY AND THROUGHPUT OF BUFFERED ALOHA WITH BACKOFF 3 II. P RELIMINARY A NALYSIS The slotted Aloha resembles the statistical multiplexer in packet switching syst ems, both of which can be considered as a system with multiple input queue s contending for a single server. The main difference lies in their contentio n resolutions . When there is more than one packet request for the output in the statistical multiplexer, one packet will be selected randomly and dispatched to t he output channel. In the slotted Aloha, however, all packets contending for the same time slot will be dismissed. In contrast to the classical infinite-node-no-buffering model [27], we will consider th e slotted Aloha as a multi-queue single-server system, as shown in Fig. 1, in which each node is equipped with an infinit e buffer and served by a single channel. Fig. 1. Multi-queue single-server system This section is devoted t o the analysis of buffered Aloha based on the decompositi on of the coupled nodes into independent Geo/G/1 queues, an approach sim ilar to the widely studied model of packet switchi ng systems [24-26]. These decomposed queues will th en be glued together by the probability of success p and the retransmission factor q of the backoff protocols. Both analyt ical and simulation results in packet switching syst ems show that this approxima tion is an effective approach with high accu racy to model the multi-queue systems [26]. A. Buffered Aloha wit h Backoff The multi-queue single-server system with n input queues can be characterized as a discrete tim e Markov process with a state space represented by the vector ( C 1 , C 2 , …, C n ) , where C i is the queue length of node i . This m ulti-dimensional Markov chain is obviously intractabl e as the number of nodes n becomes t oo large. Thus, we adopt the deco mposition approach in packet switching systems and regard each node as an independent FIFO queue with identical Bernou lli arrival processes of rate λ . The contentions are resolved by backoff rescheduling of HOL packets. The number of collisions ex perienced by an HOL packet is called the phase of the packet. Initially, a fresh HOL packet is in phase 0, and it moves to the next phase i f it is involved in a collision. Let p be the probability of a successful transmission. A K - exponenti al backoff protocol allows a packet in phase i to be transmitted with probability q i , i =0,1,…, K, where q is the retransmission factor and K is the cutoff phase of the protocol. The phase transiti on process of an HOL packet can be described by the Markov chain shown i n Fig. 2. p 1 q − 2 1 q − 1 i q − 1 K qp − 1 p − (1 ) qp − 2 (1 ) qp − (1 ) i qp − qp 2 qp i qp K qp Fig. 2. State transition diagram of HOL packets. Note that the pr obability of success p in the Markov chain is assumed t o be independent of the phase of the HOL packet. Intuitively, the chan ce that an HOL packet has a successful transmission should not vary with the num ber of collisions it has suffered. This assumption ha s been accepted and verified in various references [28]. Two particular backoff schemes are of special interest . Geometric retransmission is a special case with cutoff phase K =1, that is, the retransmission pro bability is a constant q regardless of the number of collisio ns suffered. This collision resolution algorithm is the original version of slotted Aloha protocol that has been extensivel y investigated in [1, 4-11, 18-23]. If the cutoff phase is unlimi ted, K = ∞ , then the protocol is simply called exponential backoff . For example, the binary exponential backoff (BEB) in previous studies [13-16] assumes q =1/2 and K = ∞ . Let f 0 , f 1 ,…, f K represent the limiting pro babilities of the Markov chain shown in Fig. 2. We have 1 1 (1 ) (1 ) i ii i i f fq p f q − − =− + − , i =1,…, K -1, (1) and 1 1 (1 ) (1 ) K KK K K f fq p f q p − − =− + − ( 2 ) It follows from (1-2) that 0 11 1/ 11 K qq p f pq pq p q ⎛⎞ ⎛⎞ ⎛ − ⎜⎟ =− − ⋅ ⎜⎟ ⎜ ⎜⎟ +− +− ⎝⎠ ⎝ ⎝⎠ ⎞ ⎟ ⎠ , (3) 0 1 i i p ff q ⎛ − = ⎜ ⎝⎠ ⎞ ⎟ , i =1,…, K -1, and (4) 0 1 / K K p f f q ⎛⎞ − = ⎜⎟ ⎝⎠ p . ( 5 ) Lemma 1. For buffered Aloha with K-exponential backoff (1 ≤ K ≤ ∞ ), the offered load of each queue is ρ = λ /f 0 . Proof: Since each fresh HOL packet will be in phase 0 for one time slot only, it will be either transmitted or blocked. Thus , the probability of finding a fresh HOL packet in a node buffer at any time slot, ρ f 0 , should be equal to the input rat e λ of the Bernoulli arrival process. The offered load is therefor e given by ρ = λ /f 0 , and the mean service time of each HOL packet is 1/ f 0 . □ The offered load ρ per node depends on the probability of success p and retransmission factor q, and it can be expressed as 0 11 =/ 11 K qq p f pq pq p q ρλ λ ⎛⎞ ⎛⎞ ⎛ − ⎜⎟ =− − ⋅ ⎜⎟ ⎜ ⎜⎟ +− +− ⎝⎠ ⎝ ⎝⎠ ⎞ ⎟ ⎠ ) . (6) For geometric ret ransmission ( K =1), the offered load ρ is =( 1 ) / ( ) Geo pp q p q ρλ −+ . ( 7 ) In the case of exponential backoff ( K = ∞ ), we have =/ ( 1 Exp qpq ρλ +− . ( 8 ) LEE AND DAI: STABILITY AND THROUGHPUT OF BUFFERED ALOHA WITH BACKOFF 4 Since the probability of success p is determined by the activities of all nodes in the entir e network, the offered load ρ will serve as the glue to stick separately treated q ueues together in our analysis. In the ori ginal slotted Aloha model proposed by Abramson [1], t he network throughput was obtained by assuming no buffer for queuing at each node and considering the total num ber of attempts as a Poisson random variable with parameter G. The probability of success p can then be expressed as ˆ exp( ) GG λ =− ˆ /e x p ( / ˆ ) p G λ == − p λ . In the next theorem, we will show that this characteristic of slotted Aloha still holds with an infinite bu ffer for queuing at each node, and it is invariant with respect to the retran smission factor q and the cutoff phase K under the independence assumpti on. Theorem 1. For buffered Aloha with K -exponential backoff (1 ≤ K ≤ ∞ ) , the probability of success p is given by ˆ exp ( / ) p p λ =− , ( 9 ) where ˆ n λ λ = is the aggregate input rat e. Proof: Each node in the network must be in one of the following states: State 1: idle; State 2: busy with a fresh HOL; State 3: phase i and retransmitting, i =1,2,…, K ; State 4: phase i and not retransmitting, i =1,2,…, K . We know that the probability of a node being busy is ρ , and the probability of an HOL p acket being in ph ase i given that the node is busy is f i , i =0,1,…, K . Therefore, the pr obability of th e above four states are given by: 1) Pr{node is in State 1}=1- ρ ; 2) Pr{node is in State 2}= ρ f 0 ; 3) Pr{node is in State 3}= ρ f i q i , i =1,2,… K ; 4) Pr{node is in State 4}= ρ f i (1- q i ), i =1,2,… K . When a node successfully transmits a packet, its n -1 interfering nodes m ust be either in State 1 or State 4. Suppose that the number of i dle nodes (State 1) and nodes in phase i without retransmitting (State 4) are given by m idle and m i , i =1,…, K , respectively. We have . The probability of success p in steady-state conditio ns can then be written as 1 1 K idle i i mm n = += ∑ − 1 {, , . . . , } 1 1 (1 ) ! Pr{ node is in State 1 } !! ! Pr{ node is in State 4, phase } idle idle K i m mm m idle K K m i n p mm m i = − = ⋅ ∑ ∏ " 1 1 (Pr{node is i n St ate 1 } Pr{node is i n Stat e 4, phase } ) K n i i − = =+ ∑ ( ) 1 1 1( 1 n K i i i fq ρρ − = =− + − ∑ ) ( 1 0 ) Substituting (3-6) into (10), we have () 1 1 1 (1 ) 11 n K K i i p pp p λλ − − = ⎛⎞ − =− − − ⎜ ⎝⎠ ∑ ⎟ () with a large 1 ˆ 1 / exp( / ) n n p p λ − =− ≈ − λ p G . □ According to the above theorem, the expected number of attempt s per time slot is given by ˆ /l n Gp λ == − . ( 1 1 ) It follows that the aggregate st eady-state network throughput satisfies ˆ exp( ) G λ = − , ( 1 2 ) which peaks at 1 max ˆ e λ − = when the attempt rate G =1. In spite of the fact revealed in Th eorem 1 that the probab ility of success p and the attempt rate G are independent of the retransmission factor q and the cutoff phase K , we will show that the selection of these backoff parame ters, q and K , is not arbitrary, and they are the keys to guarantee that the stable network throughput ˆ λ can be achieved by the backoff protocols. B. Stable Points of the Lambert W Function The solutions of the fundamental characteristic equation (9) of probability of success p , or it s equivalent equation (12) of attempt rate G , can be represented by the Lambert W function defined by W ( z ) e W ( z ) = z , (13) which was first considered by J. Lam bert around 1758, and later, studied by L. Euler [29] . ˆ 0.25 λ = 0 ˆ () W λ − 1 ˆ () W λ − − ˆ λ − Fig. 3. The Lam bert W function The Lambert W function is a multivalued function. If z is real and - e -1 < z <0, there are two possible real values of W ( z ): the principal branch W 0 ( z ) ∈ [-1, ∞ ] and the other branch W -1 ( z ) ∈ [- ∞ ,-1]. Both branches are illustrated in Fig. 3. The two non-zero solutions of (9) correspond, respectively , to the two branches of the Lambert W function, whose series expressions are given as follows: 1) 0 ˆ exp( ( )) L pW λ =− , and W 0 ( z ) has the following series expansion 1 23 4 5 381 2 5 0 23 2 4 1 () () ! i i i i Wz z z z z z z i − ∞ = − = =− + − + − ∑ " (14) which can be derived by using th e Lagrange i nversion theorem. According to (14), p L can be further written as 23 4 63 12 0 23 2 4 ˆˆ ˆ ˆ ˆ exp( ( )) 1 L pW λλ λ λ λ = −= − − − − − " (15) Usually 1- ˆ λ is a good approxim ation for p L as ˆ λ ≤ e -1 . 2) 1 ˆ exp( ( )) S pW λ − = − , and W -1 ( z ) has the following series expansion 23 4 5 43 769 11 1 1 3 72 540 17280 0 () 1 i i i Wz x x x x x x μ ∞ − = = =− + − + − + − ∑ " (16) in which 2( 1 ) xe z = −+ , and the coefficient μ i is given in [29]. LEE AND DAI: STABILITY AND THROUGHPUT OF BUFFERED ALOHA WITH BACKOFF 5 A good approximat ion for p S when ˆ λ > e -1 /2 is given by ( ) 2 5 2 1 33 2 ˆ exp( ( )) exp{ 2 } S pW e ˆ λ λ − =− ≈ − − + + . (17) The two non-zero roots 1 ˆ exp( ( )) S pW λ − =− and 0 ˆ exp( ( )) L pW λ =− are displayed in Fig. 4. As we can see, the roots of (9) are intersections of y = p and . In general, we have p S ≤ p L and the equality ho lds for p S = p L = e -1 when . ˆ exp( / ) y λ =− p 1 ˆ e λ − = () ˆ exp / yp λ =− ˆ 0.25 λ = 1 ˆ exp( ( )) S pW λ − =− 0 ˆ exp( ( )) L pW λ =− yp = Fig. 4. Stable points of probability of success p The previo us stability analysis based on the drifts of the number of backlogged nodes revealed that t he system has a desired stable point, an unstabl e equilibrium and an undesired stable point [6-7, 27]. The proba bility of success illustrated in Fig. 4 shows a desired stable poi nt at p L and an undesired stable point at zero, while the point p S is an unstable equilib rium point. The dynamics of these stable points characterized in the following theorem confirms the observations ma de by the drift analysis. Theorem 2 : Let p t be the probability of success at time slot t. If p t > p S then p t → p L as t →∞ . Proof: According to Theorem 1, the probability of success p t +1 at time slot t+ 1 is determined by the loading ρ t of each queue in time slot t as 1 ˆ exp( / ) tt p p λ + =− . Given p t > p S , the probab ility of success p t will converge to the desired stable point p L in the following two sub -intervals: 1. If p S < p t < p L , it can be readily seen from Fig. 4 that t 1 ˆ exp( / ) tt p pp λ + =− > t t . Therefore, p t → p L as t →∞ ; 2. If p t > p L , Fig. 4 shows that 1 ˆ exp( / ) t p pp λ + =− < . Therefore, p t → p L as t →∞ . □ We can see from Theorem 2 that the unstable equilibrium point p S is critical for the system stability. As long as the probability of success p t at time slot t stays above p S , it will ultimately converge to p L , leading to a network throughput of ˆ λ =- p L ln p L . On the other hand, if the probability of success p t drops below p S , Fig. 4 shows that it will be dep arting from the desired stable point p L because 1 ˆ exp( / ) tt t S p pp p λ + =− < < . As the probability of success becomes smaller and smaller, all the nodes will eventually become busy and the network is saturated, in which case the m ultinomial distri bution assumption adopted in the proof of Theorem 1 is no longer valid. As a consequence, the probability of success w ill not be governed by the characteristic equati on (9) displayed in Fig. 4. Instead of zero, p t will converge to an undesired stable point p A that depends on the backoff protocols. A detail ed discussion on the undesired stable point will be presen ted in Section V. As shown in Fig. 5, Theorem 2 can also be paraphrased in terms of attempt rate G t at time slot t : If G t is to the left of the unstable equilibriu m point 1 ˆ () W λ − − − , then G t converges to the desired stable point 0 ˆ () W λ − − as t →∞ . ˆ λ exp( ) GG − 0 ˆ () W λ − − 1 ˆ () W λ − −− Fig. 5. Stable points of attempt ra te G III. A BSOLUTE S TABILITY The core issue o f stability analysis is to characterize the range of retransmission fact or q , inside which the steady-state network throughput ˆ λ = n λ can be achieved. In this secti on, we will investigate the conditions of absolute stability that guarantee the convergence of pr obability of success to the desired stable point p L for achieving the throughput ˆ λ . It has been shown in Theorem 2 that the convergence of probability of success to p L requires that 1 ˆ exp{ ( )} tS pp W λ − ≥= − , ( 1 8 ) or equivalently, th e attempt rate G t satisfies 1 ˆ ln ( ) tS Gp W λ − ≤ −= − − ( 1 9 ) at any time slot t . In the next theorem, we will show that this constraint im poses an upper bound on the retransmission factor q . Theorem 3. For K-exponential backoff (1 ≤ K ≤ ∞ ) , if 1 ˆ ln / ( ) / uS qq p n W n λ − ≤= − = − − , (20) then at any time slot t , G t ≤ - ln p S . Proof: Suppose that there are totally n b backlogged HOL packets at time slot t , with n i packets in phase i , i =1,…, K . We have 1 K i i nn = b = ∑ . The attempt rate G t is then given by 1 () () K i tb i b i Gn n n q n n n λ = =− + ≤− + ∑ b q λ ( 2 1 ) where the right side of (21) is t he attempt rate corresponding to the state that the n b backlogged HOL packets are all i n phase 1. We consider th e following two cases: 1) If retransmission fact or q ≤ λ , th e attempt rate G t is bounded by LEE AND DAI: STABILITY AND THROUGHPUT OF BUFFERED ALOHA WITH BACKOFF 6 ˆ t G λ ≤ . ( 2 2 ) We know from (17) that 1 ˆ ln ( ) S pW ˆ λ λ − −= − − > . ( 2 3 ) By combi ning (22) and (23), we have ln t G ≤− S p S . (24) 2) If retransmission fact or q ≥ λ , th e attempt rate G t is bounded by ln tu Gn q n q p ≤≤ = − . ( 2 5 ) Hence, the theorem is established by com bining (24) and (25). □ Another vital criterion im posed on the range of retransmission factor q is that the offered load ρ of each input queue given by (6) must be less than 1 to ensure that t he queue length of each Geo/G/1 queue is bounded. The lower bound of the retransmission factor q , specified in the following theorem, is derived from the offered load ρ . Theorem 4. For K-exponential backoff ( 1 ≤ K ≤ ∞ ) , suppose the probability of success converges to p L , then ρ ≤ 1 iff q ≥ q l , where the lower bound q l is the root of the fol lowing equation: 1 1 1/ 11 K L LL L p qq pq pq p q λ ⎛⎞ ⎛⎞ − −− ⋅ = ⎜⎟ ⎜⎟ +− +− ⎝⎠ ⎝⎠ . (26) In particular, the low er bound q l for geometric retransmission (K=1) is ˆ (1 ) (1 ) ˆ (1 ) () Geo L L l L L p p q p pn λλ λ λ −− == − − , (27) and for exponential backoff ( K= ∞ ), we have 11 ˆ 1 1/ Exp L L l p q n λ p λ −− == − − . (28) Proof: As the probability of success p converges to p L , it is easy to show from (6) that the offered load ρ monotonical ly increases with (1- p L )/ q , and therefore ρ is a monotoni c decreasing function of q . It follows that the minimum retransmission factor q corresponds to the m aximum offered load ρ . Thus, the lower bound of q is the root of ρ =1, which t r a n s f o r m s ( 6 ) i n t o ( 2 6 ) . □ 1 1 L p λ − − (1 ) (1 ) L L p p λ λ − − ln / S pn − Fig. 6. Tradeoff between offered load ρ and retransmission factor q . Note that the retransmission factor q should be strictly larger than q l if bounded queue length is re quired. Besides, according to Theorems 2 and 3, q should not exceed q u to guarantee that the probability of success converges to p L . The offered load ρ versus the retransmission factor q is pl otted in Fig. 6 under different values of cutoff phase K . It shows that the offered load ρ is a monot onic decreasing function of q for any given K . Fig. 6 also indicates that a larger cutoff phase K leads to a higher offered load ρ , which incurs a larger delay, for any given retransmission factor q . Based on Theorems 3 and 4, we define the absolute stabl e region of retransmission factor q as . According to (20) and (27), the absolute stabl e region of geometri c retransmission i s given by [, ] lu Sq q = ˆ (1 ) ˆ () ,, l n L L p Geo Geo Geo lu S pn Sq q p λ λ − − ⎡⎤ ⎡⎤ == − ⎣⎦ ⎣⎦ / n . (29) For exponential backoff, from (20) and (28), we have 1 ˆ 1/ ,, l n L p Exp Exp Exp lu S n Sq q p λ − − ⎡⎤ ⎡⎤ == − ⎣⎦ ⎣⎦ / n . (30) If the retransmission factor is c hosen from the absolute stable region, [, ] lu qS q q ∈ = , the system will stabilize at the desired stable point p L for sure. The stable region S is determined by the aggregate input rate ˆ λ and the total number of nodes n . Apparently, the region S will becom e an empty set if either the aggregate input rate ˆ λ is too l arge or the number of nodes n goes to infinity. For any fixed n , th e ma xi mum s tab le throughput, denoted by max_ ˆ sup SS ˆ λ λ = , can be determined as follows. Lemma 2. For networks with n nodes, the maximum stable throughput is the root of the f ollowing equation: max_ ˆ S λ ˆˆ () () 0 lu qq λλ − = , (31) with retransmission f actor . max_ max_ ˆˆ () () lS u qq q λλ == S Proof: Since q l and q u are m onotonic increasing and decreasing functions, respectively, of t he aggregate input rate ˆ λ for any fixed n , the maxim um stable t hroughput must be the single root o f ( 3 1 ) . □ Note that the offered load ρ of each input queue will become 1 if the aggregate input rate i s , and the queue len gth will be unbounded. Thus, the maxim um stable throughput determined by (31) sim ply means a network throughput of max_ ˆ S λ max_ ˆ S λ max_ ˆ S λ ε − , for any positive number ε >0, is achievable with some retransm ission factor q . S ∈ The absolute stable region S Geo of geometric ret ransmission is depicted in Fig. 7. The region becom es narrower and narrower as the aggregate input rate ˆ λ increases. It eventually shrinks to a single point at , which is given in the next corollary. max_ ˆˆ Geo S λλ = LEE AND DAI: STABILITY AND THROUGHPUT OF BUFFERED ALOHA WITH BACKOFF 7 ˆ λ ˆ (1 ) ˆ () Geo L l L p q pn λ λ − = − ln / Geo uS qp n =− 1 max_ ˆ Geo S e λ − ≈ Geo S Fig. 7. Absolute stable region and m aximum stable thr oughput of geometric retransmission ( K =1). Corollary 1 . For geometric retransmission ( K=1), the maximum stable throughput is approximately given by 1 ( 3 2 ) max_ ˆ Geo S e λ − ≈ with retransmission f actor q=1/n . Proof: According to (29), the difference bet ween the upper bound and the lower bound can be approximately given by Geo u q Geo l q ( ) 1 ˆˆ ˆ () () l n ( 1 ) / Geo Geo ul S L n qq p p λλ λ −≈ − − − L p , which is a monotoni c decreasing function of ˆ λ . When ˆ λ = e -1 , the difference is , which approaches zero when the number of nodes n is large. Thus, the assertio n of this corollary directly f o l l o w s f r o m L e m m a 2 . □ 1 / en − Both the upper bound and lower bound will approach zero, and the absolute stable region of geometric retransmission will vanish, S Geo = ∅ , when the number of nodes n →∞ . As the retransmission factor q =0 is the only viable choice, a node will be unable to transmit any pack ets once it is involved in a collision. It rev eals the fact that the network cannot be stabilized at the point Geo u q Geo l q L p and the throughput is undetermined when t he number of nodes n is infinite. The absolute stable region E xp S depicted in Fig. 8 shows that the maxim um stable throughput of exponentia l backoff could be lower than e -1 . We prove in the next co rollary that it is the case for any given number of nodes n. Corollary 2 . For exponential backoff (K= ∞ ), the maximum stable throughput is approxi mately given by max_ ˆ ln / Exp S nn λ ≈ ( 3 3 ) with retransmission f actor q ≈ ln n / n . Proof: It is known from (15) that 1- ˆ λ is a good approxim ation for p L . Therefore, the lower bound E xp l q can be approxima ted by ˆ 1 ˆ ˆˆ 1/ 1/ Exp L l p q nn λ λ λλ − =≈ −− ≈ ( 3 4 ) when the number of nodes n is large. According to Lemma 2, the maxi mum throughput satisfies the following equation obtained by combining (20) and (34): max_ ˆ Exp S λ 1 ˆ () ˆ W n λ λ − − − = . ( 3 5 ) Solving (35), we can obtain (33) and the corresponding retransmission factor q ≈ ≈ ln n / n . □ max_ ˆ Exp S λ Note that the exact value of should be lower than as max_ ˆ Exp S λ ln / nn E xp l q is strictly larger than ˆ λ . Nevertheless, this approximation is quite accurate for large n . In contrast to geomet ric retransmission, here the maxim um stable throughput ≈ ln n / n decreases rapidly with the increasing number of nodes n . Again, the stable region max_ ˆ Exp S λ E xp S becomes an em pty set when the number of nodes n →∞ . In a finite-node network, however, the system can be stabilized at p L if the aggregate input rate is lower than ≈ ln n / n . This result agrees with that reported in [1 4-16], which shows that the network wit h binary exponential backoff (BEB) is stable if th e arrival rate is sufficiently small. This point is clearly illustrated in Fig. 8. It is easy to show th at if the aggregate input rate max_ ˆ Exp S λ ˆ λ is lower th an /2 1 2 n ne − , then the retransmission factor q =1/2 is included in th e stable region E xp S . ˆ λ ln / Exp uS qp n =− 1 ˆ 1/ Exp L l p q n λ − = − E xp S max_ ln ˆ Exp S n n λ ≈ /2 1 2 n ne − Fig. 8. Absolute stable region and m aximum stable thr oughput of exponential backoff ( K = ∞ ) . For the general K -exponential backoff wi th 1< K < ∞ , it is quite difficult to obtain the explicit exp ression of the lo wer bound q l . In the Appendix we show that with a large number of nodes n , the lower bound q l is approxim ately given by 1 ˆ / L l K L p q np λ − = , ( 3 6 ) and the maxim um stable throughput is 11 / 11 / max_ ˆ ln / K K S nn λ −− ≈ . ( 3 7 ) The maxim um stable throughput declines with the increasing cutoff phase K , and (37) agrees with (33) when K= ∞ . We can see from (20) and (36) that both q l and q u approach zero as the number of nodes n increases, indicating that the K -exponential backoff wit h 1< K < ∞ cannot be stabilized at the desired stable point p L either when the number of nodes i s infinite. max_ ˆ S λ The sum and substance of absolute stability is that a n etwork with K -exponential backoff can be absolutely stabilized at p L LEE AND DAI: STABILITY AND THROUGHPUT OF BUFFERED ALOHA WITH BACKOFF 8 only if the num ber of nodes n is finite. Th e maximum stable throughput could be dim inished by increasing the cutoff phase K . It appears that geom etric retransmission is a favorable option for small-scale networks because it yields the highest stable throughput e -1 . In the next section, how ever, we will sho w that absolute stability is overkill for large cutoff p hase K , and the superiority of exponenti al backoff is the existence of a larger stable region of retransm ission factor q , called the asymptotic stable region. IV. A SYMPTOTIC S TABILITY OF E XPONENTIAL B ACKOFF The absolute stable region of ret ransmission factor described in Section III guarantees that the attempt rate G t stays below –ln p S in the worst state. For exponent ial backoff, this constraint imposed on t he retransmission factor q is overly conservative, because there are an infin ite number of phases and the occurrence of the worst state that n backlogged HOL packets are all in phase 1 is an ex tremely rare event for large n . In respect to the versatility of exponential backoff with multiple retransmission phases, we consider a better-than-worst scenario th at all n nodes are backlogged with HOL packets distributed over K phases. Let n i , i =1,…, K , be the number of HOL packets in phase i , such that = n . Let 1 K i i n = ∑ i φ denote the pro bability that an HOL p acket is in p hase i given that it is backl ogged. From (3-5) we have 0 /( 1 ) ii f f φ =− , i =1,2,…, K. (38) According to the law of l arge numbers, the number of HOL packets in phase i , n i , will converge to * i nn i φ = , i =1,…, K, when the number of nodes n is large. The attempt rate, denoted by G * , under the node distribution ** 1 { , ..., } K nn can then be written as ** 11 KK i i ii Gn q n i i q φ == == ∑∑ . ( 3 9 ) An upper bound of retransmission factor that guarant ees should satisfy the fol lowing equation: * u q * ln S G ≤− p S u q * 11 ln KK ii ii ii nq n q p φ == == − ∑∑ . (40) The worst case that we have considered in Section III is that all backlogged nodes are in phase 1, corresponding to the node distributi on { n , 0, …, 0}. Thus, we can see from (40) that is no less than q u , and it provides a much looser upper bound. For geometric ret ransmission ( K =1), the upper bound is given by * u q * u q * ln / Geo uS qp n =− = , ( 4 1 ) and for exponential backoff ( K = ∞ ), we have * ln 1 Exp S uLL p qp p n =− − > u q . ( 4 2 ) For K -exponential backoff with 1< K < ∞ , the upper bound is the root of the following pol ynomial equation with degree K +1: * u q 11 1 1 11 l K LL The following approxim ation of can be obtained by using a simil ar approach given in the Appendix: * u q * ln 1 1 / S L L u p K p p q n − − − ≈ for large n . ( 4 4 ) Certainly, the upper bound determ ined by (40) does not guarantee that the system always stabilizes at the desired stable point * u q 0 ˆ exp{ ( )} L pW λ =− . The following theorem shows that the pro bability that the attempt rate G t exceeds -ln p S is insignificant, and it converges to 0 as n →∞ . Theorem 5. For exponential backoff (K= ∞ ) , if th e retransmission factor q sati sfies * ln 1 ,, 1 ˆ 1/ Exp Exp S L lu L p p qq q p p n n λ − L ⎡ ⎤ ⎡⎤ ∈= − − ⎣⎦ ⎢ ⎥ − ⎣ ⎦ , (45) then for any time slot t, th ere exists an ε > 0 such that Pr{ ln } 1 tS Gp ε ≤ −≥ − , (46) and ε→ 0 as n →∞ . Proof: Suppose that at tim e slot t there are totally n b backlogged HOL packets in the networ k, with n i packets in phase i , i =1,2…. The attempt rate G t is then given by 11 ˆ () ( 1 / ) ii tb i b ii Gn n n q n n n q λλ ∞∞ == =− + = − + i ∑ ∑ , (47) where 1 b i n ∞ = = i n ∑ . We then have 1 ˆ Pr{ l n } Pr{ ln ( 1 / ) } i tS i S b i Gp n q p n n λ ∞ = ≤− = ≤− − − ∑ 1 ε ≥− , ( 4 8 ) where 1 ˆ Pr{ ln ( 1 / ) } i iS b i nq p n n ε λ ∞ = => − − − ∑ . (49) The total number of backlogged nodes n b may vary from 0 to n . To complete the proof, we consider two cases, small n b and large n b , in the following. 1) For small n b , or specifically, () b nn ο = , according to Markov inequality we have 1 E[ ] ˆ ln ( 1 / ) i i i Sb nq p nn ε λ ∞ = ≤ −− − ∑ . ( 5 0 ) Let Y i,j be a Bernoulli random variable defined by: , 1 if backl ogged node is in phase 0o t h e r w i s e ij j i Y ⎧ = ⎨ ⎩ with the probab ility mass function Pr{ Y i,j =1}= φ i and Pr{ Y i,j =0}=1- φ i , for each backlogged node j . The number of backlogged nodes in phase i , n i , i =1,2,…. , can then be written as , 1 b n i j n = = i j Y ∑ . Therefore, the mean num ber of backlogged nodes in phase i is E[ ] ib nn i φ = . According to (50), we obtain 1 ˆ ln ( 1 / ) i bi i Sb nq p nn φ ε λ ∞ = ≤ −− − ∑ . (51) The given condition * [, ] E xp Exp lu qq q ∈ is equivalent t o 1 ˆ ln (1 ) ˆ () i S L i i L p p q n pn λ φ λ ∞ = − − ≤≤ − ∑ , ( 5 2 ) n L LL L pp qq pq pq p q p p ⎛⎞ ⎛⎞ −− −− ⋅ = + ⋅ ⎜⎟ ⎜⎟ +− +− − ⎝⎠ ⎝⎠ S n . (43) LEE AND DAI: STABILITY AND THROUGHPUT OF BUFFERED ALOHA WITH BACKOFF 9 which implies . Furthermore, we know that for given 1 (1 / ) i i i q φ ∞ = =Θ ∑ n n lim / 0 nb nn →∞ = () b n ο = . Therefore, from (50) we have 1 lim lim 0 ˆ ln ( 1 / ) i bi i nn Sb nq pn n φ ε λ ∞ = →∞ →∞ ≤ −− − ∑ = . (53) 2) For large n b , the state ( n 1 , n 2 , …) follows a multinomial distribution wit h parameters n b and ( φ 1 , φ 2 ,…), and the covariance of ( n i / n b , n j / n b ), i ≠ j , given by (/ , / ) / ib j b i j b Cov n n n n n φ φ =− , i ≠ j ( 5 4 ) becomes negligible. Accordin g to the central limit theorem, the independent random vari ables n 1 / n b , n 2 / n b , … are normally distributed. The linear combination of n i / n b , 1 1 b i i n i Z nq ∞ = = ∑ , also has a normal di stribution with mean 1 i Zi i q μ φ ∞ = = ∑ and variance 22 1 1 (1 ) b i Zi i n i q σφ φ ∞ = =− ∑ . We have ˆ (l n / ( 1 / 1 /) ) 1 Sb b Z Z pn n n λ δμ ε σ ⎛⎞ −− − + − =− Φ ⎜⎟ ⎜⎟ ⎝⎠ , (55) where δ >0 is a small positive num ber. We know from (52) that (l n ) / ZS pn μ ≤− . Besides, 22 2 11 1 11 (1 ) bb b ii Zi i i nn n ii qqq 1 i i i σ φφ φ φ ∞∞ == =− ≤ ≤ ∑∑ ∞ = ∑ . (56) Thus, the probability give n in (55) is bounded by 1 ˆ (1 / 1 / ) ( l n ) 1 bb S i i i nn n p q λ δ ε φ ∞ = ⎛⎞ −−− + ⎜ ≤− Φ ⎜⎟ ⎜ ⎝ ∑ ⎟ ⎟ ⎠ , (57) where the right side of (57) is m aximized when n b = n . Therefore, 1 1 / i i i qn δ ε φ ∞ = ⎛⎞ ⎜ ≤− Φ ⎜⎟ ⎜⎟ ⎝⎠ ∑ ⎟ n . ( 5 8 ) We have shown in (52) that . Hence, according to (58), we have 1 (1 / ) i i i q φ ∞ = =Θ ∑ 1 lim 1 lim 0 / nn i i i qn δ ε φ ∞ →∞ →∞ = ⎛⎞ ⎜ ≤− Φ = ⎜⎟ ⎜⎟ ⎝⎠ ∑ ⎟ ] . ( 5 9 ) From (53) and (59), we conclude that for any 0 ≤ n b ≤ n , ε→ 0 as n → ∞ . □ Define ** [, E xp Exp lu Sq q = as the asymptot ic stable region of exponential backoff. The m aximum asym ptotic stable throughput, denoted by * * max_ ˆ sup S S ˆ λ λ = , is given in the next corollary. Corollary 3 . For exponential backoff (K = ∞ ), the maximum asymptotic stable t hroughput is approximately given by * max_ ˆ S λ ≈ e -1 ( 6 0 ) with retransmission f actor q= 1- e -1 . Proof: The proof is similar to that of Corollary 1. According to (45), we have ln * ˆˆ () () S p Exp Exp ul n qq λλ − −≈ ⋅ L p , which is also monotonic decreasi ng with respect to ˆ λ . When ˆ λ = e -1 , the difference approaches zero when the number of nodes n i s l a r g e . □ 1 / e − n The asymptot ic stable region S * is depicted in Fig. 9 under different values of aggregate i nput rate ˆ λ . For the sake of comparison, the absolute stable region S Exp is also plotted. It can be clearly seen that S Exp is a subset of S * for any ˆ λ . Both regions are dimi nishing with an increase of ˆ λ . For , the region S Exp becomes em pty, indicating that the network cannot be absolu tely stabilized at the desired stable point p L . Nevertheless, Theorem 5 ensures that it can still be stabilized at p L with a very high pr obability, if the retransmission factor q max_ ˆˆ ln / Exp S nn λλ >≈ ∈ S * and the number of nodes n is large. ˆ λ *1 max_ ˆ S e λ − = n →∞ { } * 1 L Sp →− * ln 1 Exp s uL L p qpp n =− − 1 1 Exp L l p q λ − = − * S E xp S ln Exp s u p q n =− max_ ln ˆ Exp S n n λ ≈ 1 1 e − − ln / nn Fig. 9. Asym ptotic stable region S * as a function of aggregate input rate ˆ λ and retransmission factor q . When the number of nodes n →∞ , the absolute stable region S Exp becomes an em pty set, and the asymptotic stable region S * shrinks to a single poi nt 1- p L . The corresponding network throughput at this point can be written as ˆ ln ( 1 ) ln ( 1 ) LL p pq λ q = −= − − − , (61) where q =1- p L . The maxim um throughput e -1 is achieved when the retransmission factor q =1- e -1 . This is consistent with that claimed i n [17], which shows that the network throughput of exponential backoff with an infinite num ber of nodes is given by 1 1 ln r rr − r − , where r corresponds to 1/ q in our case. We note that for any given aggregate input rate 1 ˆ e λ − ≤ , the input rate of each node ˆ / n λλ 0 = → as n →∞ . Thus, the above result is a natural consequence o f Lemma 1 that the probability of success p L → 1- q as n →∞ is the singularity of t he mean service time 0 1/ = / = / ( 1 ) Exp L fq p q ρλ +− → ∞ (62) when the offered load ρ Exp → 1. Here, the infinite mean service time simply m eans that the ava ilability of the server for each individual queue is uncertain in the face of an infinite number of busy nodes in the network. Even though a network throughput of ˆ λ can still be achieved, the delay performance of each input queue would be severely penalized. This point will be further elaborated in the next section on the discussion of pseudo-stability. With a finite cutoff ph ase, K < ∞ , the upper bound will approach zero when the number of nodes n →∞ , indicating that * u q LEE AND DAI: STABILITY AND THROUGHPUT OF BUFFERED ALOHA WITH BACKOFF 10 no retransmission fact or q can be found to make the K -exponential backoff wi th finite K asym ptotically stable. In fact, from (44) we can see that the cutoff phase K is used to counteract the effect of the num ber of nodes n in th e K -exponential backoff schem e. With an infinite number of nodes, an infinite number of pha ses is required to stabilize the system at p L . V. P SEUDO -S TABILITY OF E XPONENTIAL B ACKOFF The preceding analysis focuses on the delineation of stable region of the retransmi ssion factor q , in which the system will stabilize at the desired stable point p L . This section is devoted to the analysis of system behavi or once the probability of success p t < p S , and the system may run into the risk of being evolved into the undesired stable point. It is proved in Theorem 2 th at the probability of success p t will monotonically decrease if p t < p S . Consequently, according to (3), the service rate f 0 of each queue will become sm aller and smaller and eventually drop below the input rate λ. In this case, the system becomes saturated a nd all nodes in the system will be busy with pr obability 1. In time slot t when all nodes are busy, for networks with geometric retransmission, all HOL packets will be in phase 1 , and the attempt rate is given by G t = nq . For exponential backoff, however, there is no limit on the phases of HOL packets. The nodes can always back off to deeper phases to alleviate contentions, and to make the attempt rate G t arbitrarily small until the network is stabilized. Thus, exponential backoff is much more adaptive to the tra ffic fluctuation when com pared with geometri c retransmission. To manifest the adaptability of various back off protocols, we will demonstrate in this secti on that there exists a non-em pty pseudo-stable region for exponential backoff, while geometric retransmission and K -exponential backoff with 1< K < ∞ are inherently unstabl e if the number of nodes is infinite. We need the following properties of th e Lambert W function to facilitate our analysis: 1. Monotonic increasing property of : 0 () Wz . (63) 1 00 1 ( ) ( ) iff Wa W b e a b − −≤ ≤ − ≤ ≤ 2. Monotonic decreasing property of : 1 () Wz − . (64) 1 11 1 ( ) ( ) iff 0 Wa W b e ab − −− −≥ ≥ − ≤ ≤ ≤ The main result of this section is o utlined in the following theorem. Theorem 6. If p t < p S at some time slot t, the probability of success p t will converge to p A as t →∞ , where p A is the root of the following equation: 1 exp / 1 11 K pq pq p pn pq pq q ⎧⎫ ⎛⎞ ⎛⎞ ⎛ − ⎪ ⎜ =− − − ⋅ ⎨ ⎜⎟ ⎜ ⎜ +− +− ⎝⎠ ⎝ ⎪⎪ ⎝ ⎩⎭ ⎞ ⎪ ⎟ ⎬ ⎟ ⎟ ⎠ ⎠ . (65) In particular, two speci al cases are of interest: 1) For geometric retransmi ssion (K=1), we have exp{ } Geo A p nq ≈− , ( 6 6 ) and as n →∞ . 0 Geo A p → 2) For exponential backof f (K= ∞ ), we have (1 ) ln( 1 ) Exp A nq p nq q − ≈ +− , ( 6 7 ) and 1 Exp A p q →− as n →∞ . Proof: Similar to the proof of Theorem 1, if all n -1 interfering nodes are busy, the probability of success at time slot t +1 can be written as ( ) with a large 1 1, 0 , 1 (1 ) e x p ( / ) n n K i ti t t i pf q n f − + = =− ≈ − ∑ t p . (68) Substituting (3) into (6 8), we have { } 1 exp / ( ) tt p ng p + =− , ( 6 9 ) where 1 () 1 11 K tt t tt pq pq p gp pq pq q ⎛⎞ ⎛ − =− − ⋅ ⎜⎟ ⎜ +− +− ⎝⎠ ⎝⎠ t ⎞ ⎟ (70) is a monotoni c decreasing function of p t . It is straightforward to show that: i) When p t -1 < p t , according to g ( p t -1 )> g ( p t ) we have p t > p t +1 ; ii) When p t -1 > p t , according to g ( p t -1 )< g ( p t ) we have p t < p t +1 . Therefore, p t converges to the unique fixed point p A of (69) as t →∞ . Two specific examples are illustrated as follows. 1) When K =1, we have ( ) exp /( 1 ) AA A p nq p p q =− − + . ( 7 1 ) Since (1- q ) p A <<1 for a large n , we have p A ≈ exp{- nq } → 0 as n →∞ . 2) When K = ∞ , we have (1 ) exp A A A qp pn pq ⎛ −− =− ⋅ ⎜ ⎝⎠ ⎞ ⎟ . ( 7 2 ) Let W = n (1/ q -1)/ p A . Then (72) can be written as . ( 7 3 ) ( 1 / 1 ) exp( / ) W We n q n q =− Since n (1/ q -1)exp( n / q )>0, the Lambert W function defined by (73) can be uniquely represented as: 0 (1 / 1) / ( (1 ) / e x p ( / ) ) A Wn q p W n q q n q = −= − ⋅ . (74) Next, apply (63) and the property W 0 ( We W )= W to the following inequality: ( n/q-n )exp( n/q-n ) ≤ ( n/q-n )exp( n/q ) ≤ ( n/q )exp( n/q ), (75) we imm ediately obtain 0 / ( (/ ) e x p (/ ) ) / nq n W nq n nq nq − ≤− ≤ . (76) This inequality allows us to assume that 0 (( / ) exp( / )) / W nq n nq nq x − =− , (77) for some 0 ≤ x ≤ n , which is the Lambert W function that satisfies: ( / ) exp( / ) ( / ) exp( / ) nq x nq x nq n nq − ⋅− = − . (78) Hence, we have exp( ) ( / ) / ( / ) x nq x nq n = −− . ( 7 9 ) Since n / q > >x for large n, from (79) we have ln( 1 ) x q ≈ −− . ( 8 0 ) Finally, we can obtain (67 ) from (74) by substitutin g (80) into ( 7 7 ) . □ In contrast to the desired stable point p L , which is invariant with respect to the retransmission factor q, the undesired stable point p A is dependent on q and the cutoff phase K . LEE AND DAI: STABILITY AND THROUGHPUT OF BUFFERED ALOHA WITH BACKOFF 11 Fig. 10 shows the probability of success p versus retransmission factor q of geometri c retransmission . It has been proven in Section III that the probability of success converges at p L with a retransmission factor . However, it drops significantly below p L when q > =-ln p S / n . As the number of nodes n →∞ , both and approach zero and the stable region S Geo becomes empty. Besides, the undesired stable point → 0 for any given ret ransmission factor q . Therefore, we can conclude that the geomet ric retransm ission system is unstable if the number of nodes is i nfinite. This result is consistent wi th the previous studies [4-5] that the slotted Aloha network with geomet ric retransmission is inherentl y unstable as n →∞ . Geo qS ∈ Geo u q Geo l q Geo u q Geo A p ˆ (1 ) ˆ () Geo L l L p q pn λ λ − = − ln Geo S u p q n − = exp( ) 0 Geo A pn q =− → Geo S =∅ Fig. 10. Probability of success p versus retransmission factor q in the geometric retransmission case ( K =1). For exponential backoff, the cu rve of probability of success p versus retransmission fact or q is shown in Fig. 11. It has been proven in Section IV that the probability of success asymptot ically converges to p L if . W e note that the probability of success almost linear ly decreases with respect to q outside the asym ptotic stable region S * . Furthermore, because the asymptoti c stable region S * → {1- p L } and the probability of success * qS ∈ ≠∅ E xp A p → 1- q , as n →∞ , the exponential backoff system m ay remain stable even when the retransmission factor q exceeds the region S * . This point will be further elaborated via the throughput anal ysis. 1 ˆ 1/ Exp L l p q n λ − = − * ln 1 Exp S uLL p qp p n =− − (1 ) ln( 1 ) Exp A nq p nq q − = +− 1 Exp A pq →− * S * {1 } L Sp →− Fig. 11. Probability of success p versus retransmission factor q in the exponential backoff case ( K = ∞ ) The stability analysis in re spect to the throughput is based on the comparison of service rate 0 f and input rate λ. Specifically, the network throughput is given by: 0 ˆˆ min { , } out nf λ λ = . (81) The network will be u nstable if 0 ˆ out nf ˆ λ λ =< , in which case some input packets will be droppe d with a finite buffer at each node, or the queue length m ay become unbounded if the buffer is infinite. Corollary 4 . For geometric retransmission (K=1) , the network throughput ˆˆ out λ λ = iff . Geo qS ∈ Proof: If , we know from Section III that the network is stable and Geo qS ∈ ˆ out ˆ λ λ = . Otherwise, we consider two cases: 1) When the retransmission factor q > =-ln p S / n , the probability of success will drop below p S and converge to Geo u q Geo A p according to Theorem 6. The corresponding service rate 0 f of each single queue satisfies: 0 exp( ) 1 Geo A Geo Geo AA np q nf nq nq pp q =≈ −+ − 1 . (82) Since nq >-ln p S ≥ 1, (82) suggests that nq can be represented by the Lambert W function - W -1 (- nf 0 ), and it follows from (64) that the given condition 10 1 ˆ () l n ( ) S Wn f n q p W λ −− − −= > − = − − ≥ (83) implies 0 ˆ nf λ < . According to (81), the network throughput is given by 0 ˆˆ exp{ } out nf nq nq λ λ = =− < . ( 8 4 ) 2) If q < , according to Theorem 4 the service rate f 0 will be lower than the input rate λ . Therefore, we have Geo l q 0 ˆˆ out nf λ λ = < . □ For exponential backoff, however, Theorem 6 shows that if the retransmissi on factor is outside the stable region, * qS ∉ , the undesired stable point E xp A p does not converge to zero even in a network with an infinit e number of nodes. The next corollary manifests t hat exponential backoff is much m ore robust than geometric retransmission when the network is saturated, and there is a stable region larger than in which a network throughput of * S 1 ˆ e λ − ≤ is achievable. Corollary 5 . For exponential backoff (K = ∞ ) , if the probability of success converges to E xp A p , we have ˆ 1 1 ˆ ˆ (1 ) l n (1 ) L S out if p q p otherwis e qq λ λ λ ⎧ −≤ ≤ − ⎪ = ⎨ −− − < ⎪ ⎩ Proof: If the probability of success converges to E xp A p , it follows from (67) that the service rate 0 f of each single queue satisfies: 0 1( 1 ) l n ( 1 ) (1 ) l n (1 ) / ln( 1 ) Exp A pq q q f qq qn q q +− −− − =≈ ≈ − − − +− n e . (85) The two roots of (85) are then given by Lam bert W function as follows: (86) 00 10 () 1 () 1 1 Wn f Wn f e q ee − − − − − ⎧ ≥ −= ⎨ ≤ ⎩ LEE AND DAI: STABILITY AND THROUGHPUT OF BUFFERED ALOHA WITH BACKOFF 12 The given condition 1- p L ≤ q ≤ 1- p S is equivalent t o 0 1 ˆ ˆ () () 1 W W SL ep q p e λ λ − − − =≤ − ≤= . (87) The combination of (86) and (87) y ields 10 0 0 0 1 ˆ ˆ () () ( ) () 1 Wn f W n f W W ee e e e λ λ − − −− − − ≤≤ ≤ ≤ − . (88) It follows from (63-64) that (88) implies 0 ˆ nf λ ≥ . According to (81), the network throughput is ˆ λ . On the other hand, if the retransmission factor satisfies either q <1- p L or 1- p S < q , using the simil ar procedure we can prove that 0 ˆ nf λ < . In this case, the network throughput is 0 ˆ out nf λ == ˆ (1 ) l n (1 ) qq λ −− − < . □ Despite its robustness, a network with exponential backoff may suffer from severe delay jitter when the probability of success drops below p S and converges to E xp A p . The nodes would have to back off to much deep er phases with extrem ely small retransmission probabilities when saturatio n occurs. As a result, once a node tries to retransmit and succeeds, it is very likely that this node will dominate the channel for a fairly long period of time and produce a continuous stream of packets until it is interrupted by the retransmission req uests initiated by other backlogged nodes. This “capture phenomenon” occurred when the network becomes saturat ed has been described in [30]. The reason that the channel coul d be captured by a single node during saturati on is twofold: After the first packet being successfully transmitted, subsequent packets in the q ueue are all in phase 0 and can be transmitted immediately once they are being moved to the HOL position. At the sam e time, the contentions from all other nodes are insignificant due to their low retransmission probabilities. The exponential backoff protocol actuall y serves as an adaptive traffic regulator of th e shared channel. When the network is saturated, a stabl e throughput is achieved by clearing the backlogged packets in one node’s queue with very little interferences from the other nodes. The channel takes random turns to serve these backlogged nodes, whose output processes may not be stationa ry any longer because of the capture effect. Fig. 12. Non-stationary queue length of a single node in a networ k with exponential backoff ( K = ∞ ), where n =50, ˆ λ =0.3 and q =0.8 . We know from Theorem 6 t hat in a saturated network with an infinite num ber of nodes n , the undesired stable point is the singularity of t he offered load ρ Exp = λ q /( p + q -1). Thus, the range {| 1 Exp A p →− q 1 1 } L S qp q p −≤ ≤ − is called the pseudo-stable region . In this region, as the probability of success drops below p S and converges to the undesired stable point E xp A p , the network throughput ˆ λ can still be achieved at the cost o f severe delay jitter, resulting unpredictable queuing behavi or. The simulation result shown in Fig. 12 demonstrat es the non-stationary queue length caused by the capture effect of a single node, which may not have a steady-state distribution. For K -exponential backoff with 1< K < ∞ , the network can yield higher throughput wi th a larger cutoff phase K outside the stable region. Similar to the case of asymptotic stability, a non-empty pseudo-stable region cannot be warranted by the K -exponential backoff wit h 1< K < ∞ as n →∞ . Sin ce the explicit expression of p A for a general K is rather complicated, we can only sketch som e distinguishing characteristics of p A in the following corollary. Corollary 6. For K-exponential backoff with 11- p S or q <1- p L mainly stems from the non-stationary queue length of each individual node when the network is saturated. exp( ) nq nq − (1 ) l n (1 ) qq − −− ˆ out λ ˆ λ Fig. 16. Network throughput ve rsus retransmission factor q with n =50 and ˆ λ = 0.3. The simulation results presented in this section reinforce the adaptability of exponential backo ff that has been investigated in Section V. It outperforms geometri c retransmission in term s of network throughput outsi de the absolute stable region when the number of nodes is large. As we have mentioned before, t he main drawback of the pseudo-stab ility attributes to the potential unbounded delay that could be e xperienced by packets waiting in the input buffers. LEE AND DAI: STABILITY AND THROUGHPUT OF BUFFERED ALOHA WITH BACKOFF 14 R EFERENCES VII. C ONCLUSIONS In this paper, a unified approach is expl oited to determine the stable region and t hroughput of a buffered Aloha network with exponential backoff collision resolution algorithms. For networks with a finite num ber of nodes, an absolute stable region can be determ ined for any K -exponential backoff that guarantees the network to converge to the desired stabl e point. Our analyses show that t he maximum throughput in this region dimini shes with increasing cutoff phase K. Thus, geometric retransmission ( K =1) could be a favorable option for small-scale net works. The requirement of absolute convergence is overkill for large cutoff phase K. We show that there exists a bigger asymptotic st able region for exponential backoff ( K = ∞ ), which guarantees th at the network will be stabilized at the desired stable poi nt with a very high probability. W hen compared to geometric retransmission, exponential backoff is more resilie nt in dealing with transient fluctuations of traffi c. For netw orks with an infi nite number of nodes, geometric ret ransmission is unstable, while exponential backoff can be pseudo-stable. [1] N. Abramson, “The Aloha System – Another Alternative for Computer Comm unication,” Proc. Fall Joint Compet. Conf. , AFIP Conference, vol. 44, pp. 281-285, 1970. [2] A. Ephr emides and B. Hajek, “Inform ation theory and communication networks: An unconsumm ated union,” IEEE Trans. In f. Theory , vol. 44, no. 6, pp. 2416-2434, Oct. 1998. [3] L. Tong, V. Naware, and P. Venkitasubramaniam , “Signal Processing in Random Access, ” IEEE Signal Processing Magazine , Sept 2004. [4] G. Fayolle, E. Gelenbe, and J. Labet oulle, “Stability and optim al control of the packet switching broadcast channle,” J. Assoc. Comput. Machinery , vol. 24, pp. 375-386, July 1977. [5] W. A. Rosenkrantz and D. Towsley, “On the instability of the slotted Aloha multiaccess algorithm,” IEEE Trans. Automat. Contr. , vol. 28, no. 10, pp. 994-996, Oct. 1983. [6] A. B. Carleial and M . E. Hellman, “Bistable behavior of Aloha-type systems,” IEEE Trans. Commun. , vol. 23, no. 4, pp. 401-410, Apr. 1975. [7] Y. –C. Jenq, “On the stability of slotted Aloha system s,” IEEE Tra ns. Commun ., vol. 28, no. 11, pp. 1936-1939, Nov. 1980. [8] S. Lam and L. Kleinrock, “Packet switching in a multi-access broadcast channel: Dynamic control pr ocedures,” IEEE Trans. Commu n. , vol. COM-23, pp. 891-904, 1975. [9] B. E. Hajek and T. va n Loon, “Decentralized dynam ic control of a multiaccess broadcast channel,” IEEE Trans. Automat. Contr. , vol. 27, pp. 559-569, June 1982. [10] V. A. Mikhailov, Methods of Random Multiple Access , Candidate Engineering Thesis, Moscow Ins titute of Physics and Technology, Moscow, 1979. Several interesting topi cs deserve future research, including the delay analysis of input queue , the compatibility of the cutoff phase K and the number of nodes n to strike a balance between network throughput and queuing dela y. Our approach can be extended to exp lore other collision reso lution pr otocols such as CSMA. [11] R. Rivest, “Network control by Bayesian broadcast,” IEEE Trans. Inf. Theory , vol. IT- 33, pp. 323-328, May 1987. [12] R. M. Metcalfe and D. R. Boggs, “E thernet: distributed packet switching for local computer networ ks,” Commun. ACM , pp. 395-404, Jul. 1976. [13] D. Aldous, “Ultimate instability of exponential back-off protocol for acknowledgement-based transm i ssion control of random access comm unication channels, ” IEEE Trans. Inf. Th eory , pp 219-223, 1987. [14] J. Goodman, A. G. Greenberg, N. Ma dras and P. March, “Stability of binary exponential backoff,” J. ACM , vol. 35, pp. 579-602, 1988. A PPENDIX . D ERIVATION OF AND OF K -E XPONENTIAL B ACKOFF (1< K < ∞ ). l q max_ ˆ S λ [15] J. Hastad, T. Leighton and B. Rogoff, “Analy sis of backoff protocols for multiple access channels,” SIAM J. Comput. , vol. 25, pp. 740-744, 1996. [16] H. AL-Ammal, L. A. Goldberg and P. MacKenzie, “An improved stability bound for binary exponential backoff,” Theory Comput. Syst. , vol. 30, pp. 229-244, 2001. The lower bound q l is the root of equation ρ =1. Let x =(1- p L )/ q . According to (6) the offered load ρ can be written as 11 1 1 1 = 11 1 K K L x K L x x xx p x p ρλ λ ⎛⎞ ⎛⎞ ⎛ − −− ⋅ = + ⎜⎟ ⎜⎟ ⎜ ⎜⎟ −− − ⎝⎠ ⎝ ⎝⎠ ⎞ ⎟ ⎠ . (94) [17] B-J Kwak, N-O Song, L. E. Miller, “Performance analysis of exponential backoff,” IEEE/ACM Trans. Networking , pp. 343-355, April 2005. [18] B. Tsybakov and W. Mikhailov, “Er godi city of Slotted Aloha System, ” Probl. Inform. Transmission , vol. 15, no. 4, pp. 73-87, 1979. [19] R. Rao and A. Ephremides, “On the stability of interacting queues in a multiple-access system ,” IEEE Tran s. Inf. Theory , pp. 918-930, 1988. Suppose that x * is the solution of equati on: [20] V. Anantharam, “The stability regi on of the finite-user slotted ALOHA protocol,” IEEE Trans. Inf. Theory , vol. 37, pp. 535-540, M ay 1991. 1 ˆ 1 K K L L xn p x x λ − += ⋅ − p . ( 9 5 ) [21] W. Szpankowski, “Stability conditions for some m ultiqueue distributed systems: Buffered random access systems,” Adv. Appl. Prob. , vol. 26, pp. 498-515, 1994. With a large n , x * >1, we have [22] W. Luo and A. Ephrem ides, “Stability of N interacting queues in random-access system s,” IEEE Trans. Inf. Theory , vol. 45, pp. 1579-1587, Jul. 1999. 1 1 K K K L x p xx x − +≈ − . ( 9 6 ) Substituting (9 6) into (95), we o btain x * approximate ly as follows [23] J. Luo and A. Ephremides, “On the throughput, capacity, and stability regions of random m ultiple access,” IEEE Trans. Inf. Th eory , vol. 52, pp. 2593-2607, June 2006. 1/ ˆ (/ ) K L xn p λ ∗ ≈ . ( 9 7 ) [24] J. Y. Hui and E. Arthurs, “A br oadband packet switch for integrated transport,” IEEE J. Select. Areas Commun. , vol. 5, pp. 1264-1273, 1987. The lower bound q l is therefore given by [25] M. J. Karol, M. G. Hluchyj, and S. P. Morgan, “Input versus output queueing on a space-divi sion packet switch,” IEEE Trans. Commun. , vo l. 35, no. 12, pp. 1347-1356, Dec. 1987. ˆ ( 1 )/ ( 1 )/ / K lL L L qp x p n p λ ∗ =− ≈− . ( 9 8 ) [26] J. Y. Hui, Switching and Traffic Theory for Integrated Broadband Networks , Kluwer Academic Publishers, 1990. When K is large, we have 11 / ˆ 1 ˆ / K L l KK K L p q nn np ˆ λ λ λ + − ≈≈ ≈ . (99) [27] D. P. Bertsekas and R. G. Gallager, Data Networks , Wiley. [28] G. Bianchi, “Performance analys is of the IEEE 802.11 distributed coordination function,” IEEE J. Select. Areas Commun. , vol. 18, no. 3, pp. 535-547, Mar. 2000. According to Lemm a 2, the maxim um stable throughput can be obtained by com bining (20) and (99): max_ ˆ S λ [29] R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey, and D. E. Knuth, “On the Lambert W function,” Adv. Comput. Math. , vol. 5, pp. 329–359, 1996. 1/ 1 1/ 1 1/ max_ ˆˆ /l n / l n / K KK SS np n n n λλ −− =− ⇒ ≈ . (100) [30] R. Rom and M. Sidi, Multiple Access Protocols: Performance and Analysis , Springer-Verlag New York I nc., 1990.