State Dependent Attempt Rate Modeling of Single Cell IEEE 802.11 WLANs
with Homogeneous Nodes and Poisson Packet Arrivals111This paper is an extended and thoroughly revised version of our earlier work Panda and Kumar (2009). In this paper, we provide new simulation results in support of the State Dependent Attempt Rate (SDAR) model of contention, and also provide complete proofs and derivations.
Analytical models for IEEE 802.11-based WLANs are invariably based on approximations, such as the well-known decoupling approximation proposed by Bianchi for modeling single cell WLANs consisting of saturated nodes. In this paper, we provide a new approach to model the situation when the nodes are not saturated. We study a State Dependent Attempt Rate (SDAR) approximation to model queues (one queue per node) served by the CSMA/CA protocol as standardized in the IEEE 802.11 DCF MAC protocol. The approximation is that, when of the queues are non-empty, the transmission attempt probability of the non-empty nodes is given by the long-term transmission attempt probability of “saturated” nodes as provided by Bianchi’s model. The SDAR approximation reduces a single cell WLAN with non-saturated nodes to a “coupled queue system”. When packets arrive to the queues according to independent Poisson processes, we provide a Markov model for the coupled queue system with SDAR service. The main contribution of this paper is to provide an analysis of the coupled queue process by studying a lower dimensional process, and by introducing a certain conditional independence approximation. We show that the SDAR model of contention provides an accurate model for the DCF MAC protocol in single cells, and report the simulation speed-ups thus obtained by our model-based simulation.
keywords:IEEE 802.11 DCF, CSMA/CA, single cell, coupled queue system, state space reduction, iterative method, model-based simulation
The IEEE 802.11 standard wan (2007) has been widely adopted as the de facto standard for accessing shared wireless media in Wireless Local Area Networks (WLANs). The Distributed Coordination Function (DCF) Medium Access Control (MAC) protocol, which is a particular version of the Carrier Sense Multiple Access with Collision Avoidance (CSMA/CA) protocols, provides the fundamental access method in 802.11 WLANs wan (2007). The Hybrid Coordination Function (HCF) which provides service differentiation and Quality of Service (QoS), and the optional Point Coordination Function (PCF) are also built on top of the DCF wan (2007). Hence, understanding the behavior of the DCF is the key to understand the performance of 802.11 based WLANs.
This paper is concerned with analytical modeling of DCF-based WLANs in the single cell scenario. As in Kumar et al. (2007), we define a single cell to be a set of closely located 802.11 nodes such that: (i) every node can sense the transmissions by every other node, and (ii) every node can decode the transmissions by every other node in absence of interference. Figure 2 depicts an example of a single cell WLAN which operates in the ad hoc mode. In this example, users or client stations (STAs) can directly communicate among themselves. Figure 2 depicts another example of a single cell WLAN which operates in the infrastructure mode. In this latter example, STAs are primarily interested in accessing the Internet through an Access Point (AP). The analytical model in this paper applies to single cells that operate either in the ad hoc mode or in the infrastructure mode. To develop our analytical model, we do not need to distinguish between the AP and the STAs, and we call an AP or a STA, a “node”. We assume that the readers are familiar with the DCF MAC protocol, the details of which can be found in the standard document wan (2007).
Analytical modeling of 802.11 WLANs has been a topic of great interest ever since the publication of the original version of the standard wan (1999). In his seminal work, Bianchi Bianchi (2000) proposed an accurate analytical model for DCF-based single cell WLANs consisting of saturated nodes. A node is said to be saturated if it always has a packet to transmit, i.e., if its transmission queue never becomes empty. The saturation model is often useful in providing a sufficient condition for stability Baccelli and Foss (1995); Kumar and Patil (1997).222It is shown in Baccelli and Foss (1995) that, if the arrival point process is a stationary and ergodic marked point process of finite intensity, then under certain constraints (which are satisfied by several classical open queueing networks), the system is stable if the intensity of the arrival process is strictly smaller than the “intensity of the departure process obtained by saturating the queues.” By “stability” here we mean that the (joint) queue length process has a proper stationary distribution. In fact, the saturation assumption adopted by Bianchi Bianchi (2000) was partly motivated by the requirement to determine the maximum traffic intensity that the system can sustain under stable conditions. The saturation assumption, of course, simplifies the modeling problem since the node queues never become empty and queueing dynamics can be ignored. However, the saturation assumption is not valid, in general. In many real applications, e.g., web browsing and packetized voice telephony, a single cell WLAN might operate well below saturation.
When a node is not saturated, its (transmission) queue can be empty for a positive fraction of time during which it does not contend for accessing the medium. Thus, queueing dynamics must be studied in order to determine the fraction of time for which a node contends. Due to the contention for medium access, however, the queue length processes of the nodes are coupled. Hence, modeling a single cell WLAN with non-saturated nodes involves analyzing a “coupled queue system” where the queues are served according to the CSMA/CA protocol as standardized in IEEE 802.11 DCF wan (2007). In this paper, we apply a State Dependent Attempt Rate (SDAR) approximation (see Approximation 3.1 in Section 3.2) to study such a coupled queue system.
1.1 Literature Survey
Bianchi Bianchi (2000) developed the seminal analytical model for DCF-based single cell WLANs consisting of saturated nodes. The key approximation in Bianchi’s model is the following:
Approximation 1.1 (Bianchi’s Approximation).
Every transmission attempt by a node collides with a constant probability regardless of the history of collisions.
Approximation 1.1 is popularly known as the decoupling approximation since it decouples the backoff processes of the nodes through a constant parameter . In reality, the backoff processes of the nodes are coupled precisely due to collisions. Applying the decoupling approximation, Approximation 1.1, Bianchi considered a tagged node, and developed a two-dimensional Markov chain to model the evolution of the backoff process of the tagged node. Since the backoff parameters (i.e., , , and the retransmit limit) of the nodes are identical in the DCF, any node can be taken as the tagged node. The two-dimensional Markov chain keeps track of the backoff stage (i.e., the number of times the same packet has already collided) and the backoff count (i.e., the remaining number of backoff slots for the next transmission attempt to begin) of the tagged node. From the stationary distribution of the two-dimensional Markov chain, Bianchi obtained “the probability that a node attempts a transmission in a randomly chosen slot” as a function of the collision probability . Let
denote this function. With saturated nodes in the single cell, Bianchi obtained the collision probability as a function of the attempt probability by
since every node attempts a transmission in any randomly chosen slot with probability . We emphasize that, is not the probability that a collision occurs in a randomly chosen slot; is the probability that an attempted transmission collides. The above two equations yield a fixed point equation . The transmission attempt probability and the system throughput could be obtained using the solution of the fixed point equation.
Cali et al. Cali et al. (2000a, b) proposed and analyzed a -persistent version of IEEE 802.11. Considering saturated nodes, they obtained a theoretical upper limit, and also proposed an adaptive backoff mechanism to achieve the theoretical throughput limit. Kumar et al. Kumar et al. (2007) generalized the Bianchi model to arbitrary distributions of backoff times, backoff multipliers and retransmit limits.
Motivated by the need to understand the performance of WLANs under realistic traffic conditions, modeling of the non-saturated case has attracted much attention in the recent past. Several models for the traffic arrival processes have been considered in the literature. Under the “Poisson traffic” model, packets of fixed size are assumed to arrive into the node queues according to independent Poisson processes of given rates Tickoo and Sikdar (2004); Cantieni et al. (2005); Malone et al. (2007); Duffy and Ganesh (2007); Bae et al. (2008); Huang and Duffy (2009); Garetto and Chiasserini (2005); Foh and Zukerman (2002). Models for packetized voice telephony have been considered in Hegde et al. (2005); Kuriakose et al. (2009). In Winands et al. (2004), the authors consider ON-OFF traffic sources with exponentially distributed OFF periods, geometrically distributed number of packets during ON periods, and exponentially distributed packet payload sizes. The Poisson, voice, and ON-OFF traffic models belong to the so-called open-loop type where the traffic sources do not adjust their sending rates depending on the level of congestion in the network. Arrivals according to the closed-loop control of TCP have been considered for long-lived flows in Kuriakose et al. (2009); Bruno et al. (2006); Miorandi et al. (2006), and for short-lived flows in Miorandi et al. (2006); Litjens et al. (2003).
As pointed out earlier, modeling the non-saturated case requires analyzing a coupled queue system. To that end, several simplifying assumptions have been made in the literature, either explicitly or implicitly. A detailed account of the veracity of common modeling hypotheses, in the context of 802.11 WLANs, can be found in Huang et al. (2008). We follow the precise terminology of Huang et al. (2008) to comment on the relevant assumptions. For any tagged node, define if the transmission attempt by the node results in a collision, and define otherwise. Also, for any tagged node, define if there is at least one packet awaiting transmission after the successful transmission from the node, and define otherwise. The simplifying assumptions in Tickoo and Sikdar (2004)-Duffy and Ganesh (2007) are then equivalent to the following assumptions regarding the sequences and :
The sequence consists of independent random variables.
The sequence consists of identically distributed random variables.
The sequence consists of independent random variables.
The sequence consists of identically distributed random variables.
In particular, the authors in Tickoo and Sikdar (2004)-Duffy and Ganesh (2007) assume that: (i) in a randomly chosen slot, each node attempts a transmission with constant probability , (ii) every (transmission) attempt collides with constant probability , and (iii) each queue is not empty after a departure with constant probability .333In general, if the nodes are not identically parametrized, e.g., in 802.11e, then Node- is associated with the probabilities , , and . However, for a given node , the probabilities , , and are assumed to be constants independent of the current state of the system. They analyze the evolution of the backoff process and the queue length process of each node in isolation and obtain fixed point equations relating , , and . The probabilities , , and are obtained by solving the fixed point equations. The collision probabilities, the throughputs and the mean packet delays are obtained using the solution of the fixed point equations.
Note that, Assumptions (A1)-(A4), essentially “decouple” the transmission attempt processes, the backoff processes and the queue length processes of the nodes through the (unknown) constant probabilities , , and , which are obtained by solving certain fixed point equations. Assumptions (A1) and (A2) regarding the collision sequence constitute the decoupling approximation introduced by Bianchi Bianchi (2000) which can be called a collision-decoupling approximation. Assumptions (A3) and (A4) regarding the queue-status sequence are specific to the non-saturated case and, together, they can be called a queue-decoupling approximation.
In Bae et al. (2008), the authors assume only (A1) and (A2) and apply a matrix-geometric analytic method. It was shown in Huang and Duffy (2009) that Assumption (A4) leads to inaccurate predictions for throughputs when the arrival rates into the queues differ significantly from each other. In Garetto and Chiasserini (2005) it is argued, by providing results from NS-2 simulations McCanne and Floyd (????), that the collision-decoupling approximation of Bianchi works well in the saturated case, but leads to inaccurate results in the non-saturated case. A detailed discussion of the validity of Assumptions (A1)-(A4) can be found in Huang et al. (2008) where the authors provide evidence from NS-2 simulations to conclude that:
Assumption (A1) is valid regardless of whether the nodes are saturated or not.
Assumption (A2) is valid when the nodes are saturated but is not valid, in general, when the nodes are not saturated.
Assumption (A3) and (A4) are not valid.
In Garetto and Chiasserini (2005), the authors propose to model the transmission attempt probability of the nodes at any instant as a function of the number of non-empty nodes in the system at . Clearly, in Garetto and Chiasserini (2005), the attempt probabilities of the nodes are state-dependent. The approach of state-dependent attempt probabilities has also been adopted in Kuriakose et al. (2009); Bruno et al. (2006). In Foh and Zukerman (2002); Miorandi et al. (2006); Litjens et al. (2003) a state-dependent service rate approach is adopted.
1.2 Preview of Contributions
We develop a new approach to model single cells with non-saturated nodes under Poisson packet arrivals. Guided by the reported inaccuracy of the state-independent approach, we adopt a state-dependent attempt probability approach as in Garetto and Chiasserini (2005); Kuriakose et al. (2009); Bruno et al. (2006). The state-dependent attempt probabilities in Garetto and Chiasserini (2005) are obtained by an iterative method which requires computations involving a three-dimensional Markov chain. We, however, apply an approximation proposed in Kuriakose et al. (2009) to obtain the state-dependent attempt probabilities (see Approximation 3.1 in Section 3.2). As explained in Section 5.2, our model is computationally less expensive than that in Garetto and Chiasserini (2005). In particular, our model requires computations involving a two-dimensional Markov chain. We emphasize that, even though we apply the approximation proposed in Kuriakose et al. (2009), the problem we address in this paper is completely different from that in Kuriakose et al. (2009). The problem setting in Kuriakose et al. (2009) is such that analysis of queueing dynamics is not required whereas we analyze a coupled queue system.
Our contributions in this paper are the following:
We develop a Markov model with Poisson packet arrivals. Our Markov model reduces a single cell WLAN with non-saturated nodes to a coupled queue system with SDAR service discipline (Section 3). We provide a sufficient condition under which the joint queue length Markov chain is positive recurrent (Theorem 3.1).
For the case when the arrival rates into the queues are equal, we propose a technique to reduce the state space of the coupled queue system (Section 4). For the case when the buffer sizes of the queues are finite and equal, we propose an iterative method to obtain the stationary distribution of the reduced state process (Section 5.1). Our iterative method is computationally less expensive than that in Garetto and Chiasserini (2005) (Section 5.2), and yet, it provides accurate predictions for important performance measures (Section 7).
We applied the SDAR approximation to modify the MAC layer of NS-2, keeping all other layers unchanged. Originally, our objective in doing so was to validate the SDAR approximation itself by comparing the results obtained from the unmodified and modified NS-2 simulations. However, by doing so, we could also demonstrate the possibility of improving the speed of simulations by model-based simulation at the MAC layer. We show that the SDAR model of contention provides an accurate model for the CSMA/CA protocol in single cells and, at the same time, achieves speed-ups (w.r.t. MAC layer operations) up to 1.55 to 5.4 depending on the arrival rates and the number of nodes in the single cell WLAN.
1.3 Outline of the Paper
We summarize our network model and assumptions in Section 2. In Section 3 we introduce the SDAR approximation and develop a Markov model with Poisson packet arrivals and infinite buffers. In Section 4, assuming equal arrival rates, we reduce the state space of the coupled queue system and obtain the transition probability matrix of the reduced state process. To demonstrate the predictive capability of our model, we restrict to the case of finite and equal buffers in Section 5. In Section 5.1 we propose an iterative method to obtain the stationary distribution of the reduced state process (for equal arrival rates, finite and equal buffers), using which, we obtain predictions for important performance measures in Section 5.3. In Section 6 we report how the SDAR heuristic technique could be applied to improve the speed of simulations. In Section 7 we validate our coupled queue model and our iterative method by comparing with NS-2 simulations where we also discuss some simulation results for the case of unequal arrival rates. Section 8 concludes the paper. Proofs and derivations have been provided in the appendices.
2 Network Model and Assumptions
We consider a IEEE 802.11 DCF-based single cell WLAN consisting of nodes. Our network model and assumptions are given in the following:
The nodes are homogeneous. This means that the nodes use identical backoff parameters, i.e., they use identical , , and “retransmit limit” (which is indeed true for the DCF MAC protocol).
The arrival processes bring packets of fixed size into the node queues according to independent Poisson processes. The arrival rate in packets/sec into Node-’s queue is denoted by .
The nodes use equal Physical layer (PHY) rates to transmit their packets.
The wireless channel is error-free which implies that single transmissions are always successful.
There is no packet capture. Simultaneous transmissions (i.e., collisions) always result in the failure of all the involved transmissions. Thus, there can be at most one successful transmission at any point of time.
3 A Coupled Queue Model
Since there can be at most one successful transmission at any point of time, the system can be viewed as a single server serving multiple queues. The queues corresponding to the nodes are coupled essentially due to MAC contention and our immediate objective is to model the evolution of the joint queue length process. We proceed by embedding at the so-called “channel slot boundaries” described in the following.
3.1 Channel Slots
Let denote the duration of a backoff slot in seconds. The duration is a PHY parameter wan (2007). We call a time unit equal to the duration of a backoff slot, a system slot. We call a period of time during which all the nodes in the system have empty queues, a system-empty period. For analytical convenience, we make the following approximations:
Nodes always sample non-zero backoffs. Consequently, when an activity period ends (i.e., after a successful transmission or a collision), subsequent transmission attempts can occur only after at least one backoff slot.
System-empty periods are integer multiples of system slots.
As per the rules of the DCF, in reality, nodes do sample 0 backoffs with some positive probability. Approximation () is also not true, in general, since arrivals occur in real time. However, it will be clear from the accuracy of our analytical model that the errors due to Approximations () and () are negligible. Owing to Approximation (), transmission attempts in the system can possibly occur only immediately after the end of a backoff slot. These possible attempt instants have been indicated by arrows in Figure 3, which depicts the backoffs and the activities in a single cell. The channel slots that occur on the common medium have also been shown. We call the time interval between any two such possible attempt instants a channel slot, and observe that the channel activity evolves over cycles of channel slots. Note that, channel slot boundaries are the only possible attempt instants.
When the system is empty, an idle channel slot of duration occurs (see Approximation ()). A succession of idle channel slots occur until arrivals make some of the nodes non-empty. Non-empty nodes sample non-zero backoffs (see Approximation ()) and attempt transmissions when their backoff counters become 0. Depending on whether there are no attempts, only one attempt, or more than one attempt made in the system, an idle, a success, or a collision channel slot occurs. The duration of an idle channel slot when the system is non-empty is equal to the duration of a backoff slot. By Approximation (), when an activity period ends, subsequent transmission attempts can only occur after a backoff slot. Hence, we combine the time duration of seconds, which immediately follows an activity period, with the activity period itself to form success or collision channel slots. The attempt process resumes at the end of channel slots, thereby creating more channel slots and the process repeats.
Let (resp. ) denote the duration of a successful transmission (resp. a collision). The success time and the collision time depend on the PHY rate of transmission. Since the nodes use equal PHY rates and the packets are of fixed size , the durations of and are fixed and equal for all the nodes. The duration corresponds to “DATA-SIFS-ACK-DIFS” in the Basic Access mode, or “RTS-SIFS-CTS-SIFS-DATA-SIFS-ACK-DIFS” in the RTS/CTS mode. The duration corresponds to “DATA-DIFS” in the Basic Access mode, or “RTS-DIFS” in the RTS/CTS mode. Note that and need not be integer multiples of system slots. The duration of a success channel slot is equal to and that of a collision channel slot is equal to .
3.2 Coupled Queue Formulation
We model the evolution of the system over discrete time instants embedded at the channel slot boundaries. Figure 4 depicts the evolution of the queue length process of a typical node in the system. Let , with , denote the channel slot boundaries. The channel slot is precisely the time interval . The duration of the channel slot is denoted by . We analyze the system in discrete time, where the discrete time index corresponds to the actual (i.e., continuous) time instant .
Let , denote the number (of packets) in the node’s queue at time . Let (resp. ), , denote the number of arrivals into (resp. departures from) the node’s queue in the channel slot. Notice the embedding of , and in Figure 4. Departures from the queues occur at the end of channel slots, since a packet is removed from a transmission queue only when an ACK is successfully received or a timeout occurs. Arrivals that occur during a channel slot are taken into account only at the next channel slot boundary, since, if a node is empty in the beginning of a channel slot, it can attempt only at the next channel slot boundary provided that packets arrive into its queue during the channel slot. The queue lengths are updated after the arrivals and the departures in the previous channel slot have been taken into account.
In this section, we assume that each node has infinite buffer space.444The infinite buffer assumption will be dropped in Section 5 and we will analyze the finite buffer case as well. Thus, , we have where . Also, , we have , and . The last constraint follows from the fact that there can be at most one successful transmission, and thus, at most one departure, in a channel slot. Clearly, the “number in the queue” processes , , evolve as:
Evidently, it must hold that , since there cannot be a departure from an empty queue.
Due to the “Poisson arrivals” assumption, the distribution of the number of arrivals in a channel slot depends only on the duration of the channel slot. The duration of a channel slot is known if the channel slot type (i.e., whether it is an idle, a success or a collision channel slot) is known (see Section 3.1). Let , and denote the duration in seconds of an idle, a success and a collision channel slot, respectively. Then, we have , and . For a given PHY, the duration is known. Also, given the packet payload size and the PHY layer (transmission) rate, the durations and can be computed Bianchi (2000); Kumar et al. (2007). With slight abuse of notation, we indicate the occurrence of an idle, a success and a collision channel slot by , and , respectively, and define, , the following probabilities:
As described in Section 3.1, nodes can attempt only at the channel slot boundaries. Only those nodes that are non-empty at can attempt a transmission at . Let denote the number of non-empty nodes in the system at . Then, by definition, we have
where denotes the indicator function. We now introduce an important approximation regarding the attempt processes of the nodes which was first proposed in Kuriakose et al. (2009).
Approximation 3.1 (State Dependent Attempt Rate).
At any channel slot boundary , , every non-empty node attempts a transmission with probability where is the attempt probability of the nodes in a single cell consisting of homogeneous and saturated nodes.
We call Approximation 3.1 the State Dependent Attempt Rate (SDAR) approximation. The ’s in the SDAR approximation can be obtained by a saturation analysis as in Bianchi (2000); Kumar et al. (2007). As a consequence of the SDAR approximation, given , the number of transmission attempts made in the system at the channel slot boundary is binomially distributed with parameters and . Hence, the probability that the channel slot is an idle, a success or a collision channel slot can be obtained as follows:
Furthermore, in case of a success channel slot, the packet departure can occur from any of the non-empty queues with equal probability. Thus, for all , , the number of departures from the queue in the channel slot satisfy
Note that, if or , then .
Owing to the channel slot structure imposed by Approximations () and (), the “Poisson arrivals” assumption, and the SDAR approximation, the joint queue length process , where
is an -dimensional Discrete Time Markov Chain (DTMC) embedded at the channel slot boundaries.
The DTMC is positive recurrent if, , we have , and
where is the aggregate throughput in packets/sec in a single cell consisting of homogeneous and saturated nodes.
Proof: See Appendix A.
The variation of with , in general, depends on the backoff parameters. However, for the default backoff parameters as prescribed in the 802.11 DCF, we observe that, for large enough ( suffices), we have, . Then, the DTMC is positive recurrent if the aggregate arrival rate is strictly less than the aggregate throughput of saturated nodes. Thus, Theorem 3.1 exhibits the connection between the saturation throughput and the maximum stable throughput of the system.
By Theorem 3.1, there exist , , such that the DTMC is positive recurrent. We denote the stationary distribution of the DTMC by , . In principle, important performance measures such as collision probability, throughput and mean packet delay can be obtained once the stationary distribution of the DTMC is known. However, the DTMC has a state space , which is difficult to handle for . The transition structure of the DTMC is such that, for , it is difficult to obtain closed-form expressions for the ’s.555The case can be easily analyzed, for example, by the “generating function approach.” Furthermore, it is not possible to numerically solve the infinite number of balance equations of the DTMC .
Thus, to validate the SDAR approximation, we replace “the detailed implementation of the IEEE 802.11 DCF in NS-2 MAC layer” with “the SDAR model of contention” keeping all other layers unchanged. The modifications are summarized in Section 6. In Section 7 we compare the simulation results obtained from (a) the unmodified NS-2 with (b) the SDAR approximation in NS-2, and show that the simulation results obtained from (a) and (b) match extremely well. This validates the SDAR approximation for infinite or finite buffers, and equal or unequal arrival rates. In Section 5.3 we obtain numerical predictions for the performance measures such as collision probability, throughput and mean packet delay for the case of equal arrival rates, equal PHY rates, and finite and equal buffers. To proceed towards that goal, we first propose a state space reduction technique in Section 4.
4 Reduction of the State Space
Suppose that the arrival rates into the queues are equal, i.e., let , . (We still retain the “infinite buffer” assumption in this section.) Thus, , , we have , , and where , , and can be obtained by substituting in Equations (2)-(4).
Definition 4.1 (Exchangeability Feller (1971)).
The random variables are said to be exchangeable if all the permutations of , where denotes a permutaion of , have the same -dimensional distribution.
Since the nodes are homogeneous and they use equal PHY rates, and since packets arrive into the node queues according to equal rate Poisson processes, it follows that, , the random variables , are exchangeable provided that the random variables , are exchangeable. We assume throughout this paper that the random variables , are exchangeable. We remark that, with unequal arrival rates into the node queues or with unequal PHY rate of transmission, the desired symmetry will be lost and the random variables , will not be exchangeable.
Since, , the random variables , are exchangeable, we consider the following alternative description of the system. We define the state of the system at a channel slot boundary by
where denotes the number (of packets) in the queue of a tagged node666Any node can be taken as the tagged node due to symmetry. at the channel slot boundary and denotes the number of nodes, other than the tagged node, that are non-empty at . Thus, by definition, we have
Note that, , we have, , and
The state space has now reduced from to .
4.1 Approximating the Process by a Process with a Few Unknown Parameters
In Appendix C we derive the one-step transition probabilities of the process where we show that the transition probabilities are functions of some unknown quantities which appear because of the following reason. To model the change in over one step, i.e., over one channel slot, one requires the probability that “a departure from a non-tagged node leaves the queue empty.” Clearly, a departure leaves the queue empty if the following two events occur together:
In the beginning of the success channel slot, the queue contains exactly one packet (which departs at the end of the success channel slot), and
The queue does not receive any packets in the success channel slot.
Event (E2) occurs with probability (see Equation (3) and recall that the arrival rates are equal). However, with the state description , the probability that event (E1) occurs cannot be determined for the non-tagged queues, since does not keep track of the number of packets in the non-tagged queues. As shown in Appendix C, the unknown quantities are precisely the probabilities , , , , given by
where , , denotes the index of a non-tagged node. Since the nodes are homogeneous, and the arrival rates and the PHY rates are equal across the node indices, the index does not really matter, and hence, does not appear in the notation . The condition indicates that there is a departure in the channel slot. The condition indicates that the departure occured from the queue. The condition must accompany the condition , since otherwise , which is a contradiction. Clearly, is the probability that the non-tagged queue from which a departure occurs in the channel slot contains exactly one packet at , given the state description .
The ’s are not known precisely because the number of packets in the non-tagged queues are not kept in the state description . Moreover, the ’s cannot be obtained from the known quantities, namely, the arrival rate and the state-dependent attempt probabilities ’s. All the other probabilities, namely, , , , , , , , , that appear in the transition probabilities of the process (see Equations (65) and (75)) can be obtained from and the ’s.
The conditions and can be eliminated from the definition of the ’s by the following lemma.
, , , , the following is true:
Proof: See Appendix B.
Lemma 4.1 says that, the probability that a non-tagged queue contains exactly packets (), given that it is non-empty and given the state description, does not depend on whether a departure occurs from that queue. Thus, applying Lemma 4.1, we redefine, , , , ,
Approximation 4.1 (Conditional Independence).
, , , , we impose the following:
Approximation 4.1 pertains to the second step in Equation (11), which amounts to saying that, the probability that a non-tagged queue contains exactly one packet, given that it is non-empty and given the number of non-empty nodes in the system, is independent of the exact number of packets in the tagged queue.
We define, , , ,
Then, Approximation 4.1 says that,
We approximate the process by a process as follows. The process has the same state description as that of . However, the transition probabilities of the process are obtained from the transition probabilities of the process by first applying Lemma 4.1 and then applying Approximation 4.1, i.e., the transition probabilities of the process now involve the unknowns , . We further impose the conditions that, , ,
where the ’s are unknown constants independent of . Thus, the process models the event (E1) through constant time-independent probabilities ’s.
With the ’s thus defined, the probability that “a departure from a non-tagged queue leaves the queue empty given that it is non-empty in the beginning of the success channel slot and that there are non-empty nodes in the system in the beginning of the success channel slot” is given by which is the joint probability of the events (E1) and (E2). Thus, we can regard the process as a DTMC (embedded at the channel slot boundaries) whose transition probabilities are functions of the unknown parameters ’s yet to be determined.
In the remainder of this paper, a random variable (resp. a quantity ) defined for the process will have an analogous random variable (resp. a quantity ) defined for the process , and vice versa. Also, random variables and probabilities without the time argument would correspond to the stationary regime assuming that the stationary regime exists.
4.2 Transition Probability Matrix of the DTMC
We define the transition probabilities of the DTMC as follows:
transition probability from the state to the state , , .
transition probability from the state to the state , , , .
The transition probabilities of the process have been derived in D (see Equations (76) and (77)). It follows from Equations (76) and (77) that each of the ’s and the ’s can be separated into two parts, a part that contains an unknown parameter and a part that does not, and we define
Let , , and denote the matrices with their entries given by , , and , respectively. Using this matrix notation, Equation (4.2) can be rewritten as
where and are diagonal matrices. Let , , , denote the stationary distribution of the DTMC (assuming that the stationary distribution exists). We define, ,
Using the above notation, the balance equations for the ’s can be written as
where the transition probability matrix has the following “ type” Neuts (1989) structure
5 The Finite Buffer Case
Let us now turn to the question of determining the unknown parameters ’s. Applying exchangeability, we can write
Equation (21) provides the clue to estimate the ’s from the stationary distribution of the DTMC as follows: ( recall that random variables and probabilities without a time argument indicate the stationary values)
where recall that , , , denote the stationary distribution of the DTMC , assuming that it exists. This suggests an iterative method of solution as follows. Given the arrival rate and the state-dependent attempt probabilities ’s, one can begin with some “guess” values for the ’s and obtain the transition probabilities. Then, the stationary state probabilities can be computed and new estimates for the ’s can be computed from the stationary distribution by applying Equation (22). This procedure can be repeated several times until the solutions converge within some specified tolerance limit. However, with infinite buffers, the DTMC would have infinite number of states and the stationary distribution cannot be computed numerically. Thus, we will apply the above idea of estimating the ’s only for the case when the nodes have finite and equal buffers so that the finite number of balance equations can be solved numerically.
Let the buffer size of each queue be packets (one for the packet in service, if any, and waiting for service in the queue). We denote the finite buffer version of the process by . (Processes, random variables and probabilities pertaining to the finite buffer case will be denoted by adding a superscript .) Recall that we interpreted the process as a DTMC embedded at the channel slot boundaries whose transition probabilities are functions of the unknown parameters ’s. Similarly, we interpret the process as a DTMC embedded at the channel slot boundaries whose transition probabilities are functions of the unknown parameters ’s. Let , , , denote the stationary distribution of the DTMC (assuming that the stationary distribution exists).
We define, ,
Then the balance equations for the finite buffer case can be written as
where the transition probability matrix has the following structure
The ’s and the ’s in are given by Equation (18), except that and need to be replaced by