Performance Analysis of IEEE 802.11 DCF under Nonsaturation Condition

Abstract

Carrier sense multiple access with collision avoidance (CSMA/CA) methods are considered to be attractive MAC protocols for wireless LANs. IEEE 802.11 distributed coordination function (DCF) is a random channel access scheme based on CSMA/CA method and the binary slotted exponential backoff procedure to reduce the packet collision. In this paper, we propose a new analytical model for a nonsaturated IEEE 802.11 DCF network and evaluate its performance. We verify our model using simulations and show that our results agree with the simulations.

1. Introduction

In recent years, a rapid evolution in wireless local area networks (WLANs) has been witnessed. IEEE 802.11-based medium access control (MAC) protocols have been widely used for WLANs. The IEEE 802.11 MAC defines the contention-based distributed coordination function (DCF) [1, 2]. In order to prevent the interference and confirm a successful transmission, the DCF includes two access techniques: basic and request-to-send/clear-to-send (RTS/CTS) access mechanisms. The basic access mechanism is a two-way handshaking method where the transmitter transmits a data frame and the receiver replies with an acknowledgment (ACK) frame to confirm a successful transmission. In addition to the basic access method, the RTS/CTS mechanism reserves the medium before transmitting a data frame by transmitting an RTS frame and replying with a CTS frame.

Many works on the modeling of IEEE 802.11 DCF have been studied [3–11]. Bianchi [3] was the first to derive a model that incorporates the exponential backoff process inherent to IEEE 802.11 as a bidimensional Markov chain. The bidimensional Markov chain model has become a common method to study the performance of the IEEE 802.11 MAC protocol and its enhancements. In [5–7], the backoff time was assumed to be geometrically distributed with a parameter related to the contention status of the medium. Wu et al. [9] followed the same Markov chain model and considered packet retransmission limits to avoid overestimating the throughput of 802.11 as in [3]. However, in most of the analytical papers in the literature, analyses have been conducted under saturation environment, that is, each station has at least one packet to transmit at each time. There is some previous modeling work for nonsaturation environment. For nonsaturation analysis, Ziouva and Antonakopoulos [12] presented an extension of the saturated Bianchi model [4] for nonsaturation analysis. In the analysis, a station is assumed to be empty immediately after the station starts to transmit a packet. In [13], the authors extended [4] for the nonsaturation traffic condition with post-backoff. In the analysis, the MAC buffer is assumed to be always empty when a buffered packet enters a backoff procedure. This assumption is not realistic even when the traffic load is very low and not bursty. Furthermore, the fact that the distributed interframe space (DIFS) is not the integral multiples of an idle slot time was not taken into account. Tickoo and Sikdar [14] developed a simple model using parameters from the saturated model, but the results appear to exhibit poor accuracy. The model in [15] attempts to integrate Bianchi's model with a queueing model, and requires a solution of a fixed-point equation spanning parameters of both the Markov chain and the queueing model. Alizadeh-Shabdiz and Subramaniam [16] extended the Markov chain of [4] to obtain an M/G/1 queueing model for nonsaturation case. Liaw et al. [17] introduced an idle state, not present in Bianchi's model [4], accounting for the case in which the station buffer is empty after a successful completion of a packet transmission. The probability that there is at least a packet in the buffer after a successful transmission is assumed to be constant and independent of the access delay of the transmitted packet. Based on the Markovian state transition model proposed by Liaw et al. [17], Daneshgaran et al. [18] proposed a linear model of the throughput of the IEEE 802.11 DCF protocol under nonsaturation traffic conditions.

In this paper, we propose a new mathematical model to study nonsaturation performance of the DCF more accurately, and then we show the accuracy of our model via computer simulations. Even though the post-backoff procedure is not considered in our proposed analysis model, our approach can be justified by the mathematical tractability of the problem.

2. Backoff Procedure in IEEE 802.11 DCF

The DCF is the fundamental access method of the IEEE 802.11. It is based on the CSMA/CA and a backoff procedure to reduce the collision probability between multiple stations accessing the channel. The CSMA/CA mechanism defines two channel states: idle and busy. If a station senses no transmission on the channel, it considers the channel state as idle; otherwise it considers the channel state as busy. In this section, we focus on the basic access mechanism of IEEE 802.11 DCF. Suppose that there are no pending packets in a station. When the station has first a packet to transmit, it starts its carrier sensing to determine the current state of the channel. If the channel is sensed as idle for a period of time equal to DIFS, the station transmits the first arrived packet immediately. If not, the station must wait until the channel becomes idle and subsequently remains idle for a DIFS period. After that the station has to wait a random backoff interval before it is permitted to transmit its packet. The remaining backoff interval is indicated by the backoff counter of the station. Initially, the backoff counter is uniformly chosen in a range [0,CW_min−1], where CW_min is known as the minimum contention window size. Whenever the station senses channel as idle during a slot time, the backoff counter is decremented by one. If the channel is sensed as busy, the backoff counter is frozen. After the channel becomes idle again for a period of DIFS, the station resumes the decrement of the backoff counter. When the backoff counter reaches zero, the station transmits its packet and waits for reception of an ACK from the destination after a time interval called short interframe space (SIFS). When the station receives the ACK within the ACK timeout interval, it will immediately perform a backoff procedure, known as post-backoff, even though no additional packets are queued. If the transmission fails, the station doubles the contention window size up to a predefined maximum value CW_max and repeats the backoff procedure.

3. Markov Chain Model

We assume the following conditions: (1) a single hop WLAN in which all stations are in the transmission range of each other such that we do not have any hidden terminal, (2) no post-backoff procedure, and (3) ideal channel condition, that is, no capture effect. Since channel conditions are assumed to be ideal, transmission errors are a result of packet collision only. Also, we assume that all stations are identical with respect to their parameters and arrival rates, and that there is no retransmission limit.

To analyze the nonsaturation performance of DCF, we observe the state of a given tagged station at the following 3 types of embedded points (see Figure 1).

Figure 1 

Embedded points.

Type 1:the end of each time period during which the channel is busy due to packet transmissions, regardless of success or collision, if the tagged station is empty at that time.Type 2:DIFS after each time period during which the channel is busy due to packet transmission, regardless of success or collision.Type 3:every slot times after the embedded points of type 2 until the channel is sensed to be busy. We assume that the duration of DIFS is equal to 2𝜎+𝜎₂ [1], where 𝜎 denotes one slot time and 𝜎₂ denotes the duration of SIFS. For notational convenience, we denote SIFS and DIFS as the duration of SIFS and DIFS, respectively.

At each embedded point 𝑡, we define a stochastic process 𝑠(𝑡) as follows:𝑠(𝑡)=−2if𝑡isanembeddedtype1;−1if𝑡isanembeddedpointoftype2ortype3atwhichthebackoffcounterofthetaggedstationisnotactivatedyet;𝑖if𝑡isanembeddedpointatwhichthebackoffstageis𝑖for0≤𝑖≤𝑚,where𝑚isthemaximumbackoffstage. (3.1)Since we do not consider post-backoff procedure, the backoff counter of a station is inactivated when the station becomes empty, and reactivated DIFS after the time period during which the channel is sensed to be busy. Let 𝑏(𝑡) indicate the following:

(1)when 𝑠(𝑡)=−2, 𝑏(𝑡)=0;(2) when 𝑠(𝑡)=−1,𝑏(𝑡)=0if2𝜎≤𝑡_𝑝<DIFS,1ifDIFS−𝜎≤𝑡_𝑝<2𝜎,2if𝜎≤𝑡_𝑝<DIFS−𝜎,3ifDIFS−2𝜎≤𝑡_𝑝<𝜎,4if0<𝑡_𝑝<DIFS−2𝜎,5if𝑡_𝑝=0, (3.2)where 𝑡_𝑝 denotes the time period elapsed since the first packet arrived at the tagged station which was empty;(3)when 0≤𝑠(𝑡)≤𝑚, 𝑏(𝑡) represents the value of the backoff counter at 𝑡. Note that, when 𝑠(𝑡)=−1, the value 𝑏(𝑡) indicates the time at which the first packet enters an empty station. An empty station waits until a packet enters its buffer. Upon the entrance of the packet, the station starts its carrier sensing to determine the current state of the channel. If the channel is idle for a period of time DIFS, the station transmits the packet without experiencing any further backoff procedure. Thus, the transmission epoch of the first packet is affected by the time the packet enters the station. For example, when 𝑠(𝑡)=−1 and 𝑏(𝑡)=0, the packet may be transmitted during the first 𝜎₂ period of the next slot; when 𝑠(𝑡)=−1 and 𝑏(𝑡)=1, the packet may be transmitted during the last 𝜎₁ period of the next slot, where 𝜎₁=𝜎−SIFS; when 𝑠(𝑡)=−1 and 𝑏(𝑡)=2 or 3, the packet may be transmitted during the next slot after one; when 𝑠(𝑡)=−1 and 𝑏(𝑡)=4, the packet may be transmitted during the first 𝜎₂ period of the next slot after two (see Figure 2). The affection is deeper in light traffic conditions than in heavy traffic conditions.

Figure 2 

Timing diagram of the embedded points.

Figure 3 

State transition diagram of the proposed IEEE 802.11 DCF model.

At embedded point 𝑡, the state of the tagged station is defined by (𝑠(𝑡),𝑏(𝑡)). For example, the state (−2,0) represents that the tagged station does not have any packet to transmit immediately after the transmission of a packet at any station in the network. The state (−1,𝑗), 𝑗=0,1,…,5, represents that the backoff procedure is not activated yet, where 𝑗 indicates when the first packet arrived after the state (−2,0). For 𝑖=0,1,…,𝑚, the state (𝑖,𝑗) denotes the backoff state, where the value 𝑖 is the backoff stage, the value 𝑗, 0≤𝑗≤CW_𝑖, is the possible backoff counter value at the backoff stage 𝑖, and CW_𝑖 is the size of contention window at stage 𝑖.

The stochastic process {(𝑠(𝑡),𝑏(𝑡))} constitutes a 2-dimensional Markov chain, which models the tagged station. The state transition diagram of the Markov chain of the tagged station is presented in Figure 3, where we use the following notations.

(1)𝛼₁, 𝛼₂, 𝛼₃, 𝛼₄, and 𝛼₅: probabilities that, after the state of a station becomes (−2,0), the first packet arrives at the empty station between 0 and 𝜎₂, between 𝜎₂ and 𝜎, between 𝜎 and 𝜎+𝜎₂, between 𝜎+𝜎₂ and 2𝜎, and between 2𝜎 and DIFS, respectively. Note that, because the time period between an embedded point with state (−2,0) and the next embedded point is always DIFS, the elapsed time 𝑡_𝑝 after the first packet arrivals is 2𝜎≤𝑡_𝑝<DIFS with probability 𝛼₁, DIFS−𝜎≤𝑡_𝑝<2𝜎 with probability 𝛼₂, 𝜎≤𝑡_𝑝<DIFS−𝜎 with probability 𝛼₃, DIFS−2𝜎≤𝑡_𝑝<𝜎 with probability 𝛼₄, and 0<𝑡_𝑝<DIFS−2𝜎 with probability 𝛼₅ at the next embedded point, at which the state of the station becomes (−1,0), (−1,1), (−1,2), (−1,3), and (−1,4), respectively (see Figure 2). For example, when packets arrive at the station according to a Poisson process with rate 𝜆, we have𝛼₁=1−𝑒^−𝜆𝜎2,𝛼₂=𝑒^−𝜆𝜎2−𝑒^−𝜆𝜎,𝛼₃=𝑒^−𝜆𝜎−𝑒^{−𝜆(𝜎+𝜎2)},𝛼₄=𝑒^{−𝜆(𝜎+𝜎2)}−𝑒^−2𝜆𝜎,𝛼₅=𝑒^−2𝜆𝜎−𝑒^−𝜆DIFS. (3.3)(2)𝛽₁ (resp., 𝛽₂): probability that, after the state of a station becomes (−1,5), the first packet arrives at the empty station between 0 and 𝜎₁ (resp., 𝜎₁ and 𝜎). When packets arrive at the station according to a Poisson process with rate 𝜆, we have𝛽₁=1−𝑒^−𝜆𝜎1,𝛽₂=𝑒^−𝜆𝜎1−𝑒^−𝜆𝜎. (3.4)(3)𝑎₀(𝑙): probability that there are no packet arrivals into a station for time period 𝑙. When packets arrive at the station according to a Poisson process with rate 𝜆, we have𝑎₀(𝑙)=𝑒^−𝜆𝑙. (3.5)For analytical simplicity, the probability 𝑎₀(𝑙) is approximated as 𝑎₀(𝐸[𝑙]) in this section.(4)𝑜_𝑙: random variable representing the time period until the first packet arrives, given that at least one packet arrives at a station for time period 𝑙. Let 𝑜_𝑙,𝑘, 𝑘≥1, be the random variable representing the time period until the first packet arrives, given that there are 𝑘 arrivals for time period 𝑙. When packets arrive at the station according to a Poisson process with rate 𝜆, the random variable 𝑜_𝑙,𝑘 is the first-order statistic corresponding to a random sample of size 𝑘 from a uniform distribution over the interval [0,𝑙], and its mean is 𝑙/(𝑘+1) [19]. Hence,𝑜_𝑙≡𝐸[𝑜_𝑙]=^∞_𝑘=1𝑙𝑘+1(𝜆𝑙)^𝑘𝑒^−𝜆𝑙𝑘!(1−𝑒^−𝜆𝑙)=1−(1+𝜆𝑙)𝑒^−𝜆𝑙𝜆(1−𝑒^−𝜆𝑙). (3.6)(5)𝑝: probability that, from the tagged station's point of view, at least one of the other stations transmits a packet at the beginning of a slot time.(6)𝑝^′ (resp., 𝑝^′): conditional probability that one of the stations except the tagged station transmits a packet during the first 𝜎₂ period of a slot time, given that the tagged station has a packet (resp., no packets) available for transmission during the time period and no stations transmit a packet at the beginning of the slot time.(7)𝑝^′′ (resp., 𝑝^′′): conditional probability that one of the stations except the tagged station transmits a packet during the last 𝜎₁ period of a slot time, given that the tagged station has a packet (resp., no packets) available for transmission during the time period and no stations transmit a packet at the beginning of the slot time and during the first 𝜎₂ period of the slot time.(8)𝜏: probability that a station transmits a packet at the beginning of a slot time.(9)𝜏^′: probability that a station has a packet available for transmission during the first 𝜎₂ period of a slot time, given that the station did not transmit a packet at the beginning of the slot time.(10)𝜏^′′: probability that a station has a packet available for transmission during the last 𝜎₁ period of a slot time, given that the station did not transmit a packet at the beginning of the slot time and during the first 𝜎₂ period of the slot time.(11)𝑑_𝑖: random variable representing the time period from the beginning of stage 0 at a station to the completion of a successful transmission from the station, given that the transmission occurs at stage 𝑖.(12)𝑃_empty: probability that a station has only one packet when the backoff stage of the station becomes 0. Since the probability that there are no packet arrivals during the time interval between when a packet becomes the first packet in the buffer and when the packet's transmission is finished is zero under saturation traffic condition, that is, 𝑎₀(𝑑_𝑖)=0 for 𝑖=0,1,…,𝑚, the states (−2,0) and (−1,𝑗) of the Markov chain {(𝑠(𝑡),𝑏(𝑡))} are transient states under saturation traffic condition. Thus, the steady-state probabilities of the Markov chain {(𝑠(𝑡),𝑏(𝑡))} under saturation traffic condition are the same as those of Bianch's model [4]. Hence, this paper focuses on the stations under saturation traffic conditions.

We consider the number of contending stations as fixed, defined as 𝑛. Let 𝑙_𝑖 be the time period between when a station enters stage 𝑖 and when the backoff counter of the station becomes 0 at stage 𝑖. Note that the backoff counter decreases by one for each idle time slot and is suspended when the channel is busy. We consider the mean time that elapses for one decrement for the backoff counter: the probability that at least one of the channels except the tagged station transmits a packet at the beginning of a slot time is 𝑝, and the mean elapsed time for one decrement in this case is 𝑇_𝑏+𝜎, where 𝑇_𝑏 denotes the required time for the freezing time due to the busy channel. The time period 𝑇_𝑏 is obtained as𝑇_𝑏=(𝑛−1)𝜏(1−𝜏)^𝑛−2𝑝𝑇_𝑠+[1−(𝑛−1)𝜏(1−𝜏)^𝑛−2𝑝]𝑇_𝑐, (3.7)where 𝑇_𝑠 denotes the required time for the successful transmission of a packet and 𝑇_𝑐 is the wasting time due to the collision of a transmitted packet. The probability that one of the channels except the tagged station transmits a packet during the first 𝜎₂ period of a slot time is (1−𝑝)𝑝^′, and the mean elapsed time in this case is approximated as 𝑜_𝜎2+𝑇_𝑠+𝜎. The probability that one of the channels except the tagged station transmits a packet during the last 𝜎₁ period of a slot time is (1−𝑝)(1−𝑝^′)𝑝^′′, and the mean elapsed time in this case is approximated as 𝜎₂+𝑜_𝜎1+𝑇_𝑠+𝜎. Thus, the mean 𝑙_𝑖≡𝐸[𝑙_𝑖] of 𝑙_𝑖 can be obtained as𝑙_𝑖=CW_𝑖2×[𝑝(𝑇_𝑏+𝜎)+(1−𝑝)𝑝^′(𝑜_𝜎2+𝑇_𝑠+𝜎)+(1−𝑝)(1−𝑝^′)𝑝^′′(𝜎₂+𝑜_𝜎1+𝑇_𝑠+𝜎)+(1−𝑝)(1−𝑝^′)(1−𝑝^′′=)𝜎]CW_𝑖2×[𝜎+𝑝𝑇_𝑏+(1−𝑝)𝑝^′(𝑜_𝜎2+𝑇_𝑠)+(1−𝑝)(1−𝑝^′)𝑝^′′(𝜎₂+𝑜_𝜎1+𝑇_𝑠)]. (3.8)Using 𝑙_𝑖, the mean 𝑑_𝑖 of 𝑑_𝑖 can be obtained as𝑑_𝑖=𝑇_𝑠−DIFS+𝑖𝑇_𝑐+^𝑖_𝑘=0𝑙_𝑘 (3.9)for 0≤𝑖<𝑚; assuming no retransmission limit, we obtain𝑑_𝑚=𝑇_𝑠𝑝−DIFS+(𝑚+1−𝑝)𝑇_𝑐+^𝑚−1_𝑘=0𝑙_𝑘+𝑙_𝑚1−𝑝. (3.10)

4. Mathematical Analysis

Let 𝑏_𝑖,𝑗 be the stationary probability of the described Markov chain. The stationary probabilities satisfy the following balance equations:𝑏_−1,0=𝛼₁𝑏_−2,0+(1−𝑝)(1−𝑝^′)(1−𝑝^′′)𝑏_−1,2;𝑏_−1,1=𝛼₂𝑏_−2,0+(1−𝑝)(1−𝑝^′)(1−𝑝^′′)𝑏_−1,3;𝑏_−1,2=𝛼₃𝑏_−2,0+(1−𝑝)(1−𝑝^′)(1−𝑝^′′)𝑏_−1,4;𝑏_−1,3=𝛼₄𝑏_−2,0+(1−𝑝)(1−𝑝^′)(1−𝑝^′′)𝛽₁𝑏_−1,5;𝑏_−1,4=𝛼₅𝑏_−2,0+(1−𝑝)(1−𝑝^′)(1−𝑝^′′)𝛽₂𝑏_−1,5;𝑏_−1,5=𝑎₀(DIFS)𝑏_−2,0+(1−𝑝)(1−𝑝^′)(1−𝑝^′′)𝑎₀(𝜎)𝑏_−1,5;𝑏_𝑖,0=𝑝^𝑖𝑏_0,0𝑏,1≤𝑖<𝑚;_𝑚,0=𝑝^𝑚𝑏1−𝑝^0,0;𝑏_𝑖,𝑘=CW_𝑖+1−𝑘CW_𝑖𝑏+1^𝑖,0,0≤𝑖≤𝑚,0≤𝑘≤CW_𝑖. (4.1)From state transition diagram, the expression for 𝑝_0,0 is given by𝑏_0,0={𝑝+(1−𝑝)𝑝^′+(1−𝑝)(1−𝑝^′)(1−𝑎₀(𝑇_𝑠))}𝑏_−1,0+{𝑝+(1−𝑝)𝑝^′+(1−𝑝)(1−𝑝^′)𝑝^′′+(1−𝑝)(1−𝑝^′)(1−𝑝^′′)(1−𝑎₀(𝑇_𝑠))}𝑏_−1,1+{𝑝+(1−𝑝)𝑝^′+(1−𝑝)(1−𝑝^′)𝑝^′′}(𝑏_−1,2+𝑏_−1,3+𝑏_−1,4)+{𝑝(1−𝑎₀(𝑇_𝑏−DIFS))+(1−𝑝)𝑝^′(1−𝑎₀(𝑜_𝜎2+𝑇_𝑠−DIFS))+(1−𝑝)(1−𝑝^′)𝑝^′′(1−𝑎₀(𝜎₂+𝑜_𝜎1+𝑇_𝑠−DIFS))}𝑏_−1,5+^𝑚_𝑖=0(1−𝑝)(1−𝑃_empty𝑎₀(𝑑_𝑖))𝑝_𝑖,0={𝑝+(1−𝑝)𝑝^′+(1−𝑝)(1−𝑝^′)(1−𝑎₀(𝑇_𝑠))}𝑏_−1,0+{𝑝+(1−𝑝)𝑝^′+(1−𝑝)(1−𝑝^′)𝑝^′′+(1−𝑝)(1−𝑝^′)(1−𝑝^′′)(1−𝑎₀(𝑇_𝑠))}𝑏_−1,1+{𝑝+(1−𝑝)𝑝^′+(1−𝑝)(1−𝑝^′)𝑝^′′}(𝑏_−1,2+𝑏_−1,3+𝑏_−1,4)+{𝑝(1−𝑎₀(𝑇_𝑏−DIFS))+(1−𝑝)𝑝^′(1−𝑎₀(𝑜_𝜎2+𝑇_𝑠−DIFS))+(1−𝑝)(1−𝑝^′)𝑝^′′(1−𝑎₀(𝜎₂+𝑜_𝜎1+𝑇_𝑠−DIFS))}𝑏_−1,5+𝑏_0,0−(1−𝑝)𝑃_empty[^𝑚−1_𝑖=0𝑎₀(𝑑_𝑖)𝑝^𝑖+𝑝^𝑚𝑎1−𝑝⁰(𝑑_𝑚)]𝑏_0,0. (4.2)Thus,𝑏_0,0=[{𝑝+(1−𝑝)𝑝^′+(1−𝑝)(1−𝑝^′)(1−𝑎₀(𝑇_𝑠))}𝑏_−1,0+{𝑝+(1−𝑝)𝑝^′+(1−𝑝)(1−𝑝^′)𝑝^′′+(1−𝑝)(1−𝑝^′)(1−𝑝^′′)(1−𝑎₀(𝑇_𝑠))}𝑏_−1,1+{𝑝+(1−𝑝)𝑝^′+(1−𝑝)(1−𝑝^′)𝑝^′′}(𝑏_−1,2+𝑏_−1,3+𝑏_−1,4)+{𝑝(1−𝑎₀(𝑇_𝑏−DIFS))+(1−𝑝)𝑝^′(1−𝑎₀(𝑜_𝜎2+𝑇_𝑠−DIFS))+(1−𝑝)(1−𝑝^′)𝑝^′′(1−𝑎₀(𝜎₂+𝑜_𝜎1+𝑇_𝑠−DIFS))}𝑏_−1,5]×1(1−𝑝)𝑃_empty[∑^𝑚−1_𝑖=0𝑎₀(𝑑_𝑖)𝑝^𝑖+(𝑝^𝑚/(1−𝑝))𝑎₀(𝑑_𝑚.)] (4.3)Hence, each of the stationary probabilities can be expressed in terms of 𝑏_−2,0, which can be obtained from the normalization condition𝑏_−2,0+⁵_𝑘=0𝑏_−1,𝑘+^𝑚_𝑖=0^CW𝑖_𝑘=0𝑏_𝑖,𝑘=1. (4.4)

Each of 𝜏, 𝜏^′, and 𝜏^′′ can be expressed as a function of the stationary probabilities. Since the probability that an embedded point is the beginning of a slot time is 1−𝑏_−2,0 and the probability that a station transmits a packet at the embedded time is ∑^𝑚_𝑖=0𝑏_𝑖,0, the probability 𝜏 is given by∑𝜏=^𝑚𝑖=0𝑏_𝑖,01−𝑏_−2,0. (4.5)Since (1−𝑟)𝑟^′ is the probability that a station has a packet available for transmission during the first 𝜎₂ period of a slot time and that is given by 𝑏_−1,0/(1−𝑏_−2,0), the probability 𝜏^′ is given by𝜏^′=𝑏_−1,0(1−𝑏_−2,0)(1−𝜏). (4.6)Since (1−𝜏)(1−𝜏^′)𝜏^′′ is the probability that a station has a packet available for transmission during the last 𝜎₁ period of a slot time and that is given by 𝑏_−1,1/(1−𝑏_−2,0), the probability 𝜏^′′ is given by𝜏^′′=𝑏_−1,1(1−𝑏_−2,0)(1−𝜏)(1−𝜏^′). (4.7)

Having obtained 𝜏, 𝜏^′, and 𝜏^′′, the probabilities 𝑝, 𝑝^′, 𝑝^′, 𝑝^′′, and 𝑝^′′ can be determined as follows:𝑝=1−(1−𝜏)^𝑛−1,𝑝^′=^𝑛−1_𝑖=1𝑖𝑖+1𝑖𝑛−1(𝜏^′)^𝑖(1−𝜏^′)^{𝑛−𝑖−1},𝑝^′=1−(1−𝜏^′)^𝑛−1,𝑝^′′=^𝑛−1_𝑖=1𝑖𝑖+1𝑖𝑛−1(𝜏^′′)^𝑖(1−𝜏^′′)^{𝑛−𝑖−1},𝑝^′′=1−(1−𝜏^′′)^𝑛−1. (4.8)

Let ℎ_𝑖,𝑗 denote the mean sojourn time at state (𝑖,𝑗). Then,ℎ_−2,0ℎ=DIFS,_−1,0=𝑝𝑇_𝑏+(1−𝑝)𝑝^′(𝑜_𝜎2+𝑇_𝑠)+(1−𝑝)(1−𝑝^′)𝑎₀(𝑇_𝑠)(𝑜_𝜎2+𝑇_𝑠−DIFS)+(1−𝑝)(1−𝑝^′)(1−𝑎₀(𝑇_𝑠))(𝑜_𝜎2+𝑇_𝑠ℎ),_−1,1=𝑝𝑇_𝑏+(1−𝑝)𝑝^′(𝑜_𝜎2+𝑇_𝑠)+(1−𝑝)(1−𝑝^′)𝑝^′′(𝜎₂+𝑜_𝜎1+𝑇_𝑠)+(1−𝑝)(1−𝑝^′)(1−𝑝^′′)𝑎₀(𝑇_𝑠)(𝜎₂+𝑜_𝜎1+𝑇_𝑠−DIFS)+(1−𝑝)(1−𝑝^′)(1−𝑝^′′)(1−𝑎₀(𝑇_𝑠))(𝜎₂+𝑜_𝜎1+𝑇_𝑠ℎ),_−1,2=𝑝𝑇_𝑏+(1−𝑝)𝑝^′(𝑜_𝜎2+𝑇_𝑠)+(1−𝑝)(1−𝑝^′)𝑝^′′(𝜎₂+𝑜_𝜎1+𝑇_𝑠)+(1−𝑝)(1−𝑝^′)(1−𝑝^′′ℎ)𝜎,_−1,𝑘=ℎ_−1,2ℎ,𝑘=3,4,_−1,5=𝑝𝑎₀(𝑇_𝑏−DIFS)[𝑇_𝑏−DIFS]+𝑝[1−𝑎₀(𝑇_𝑏−DIFS)]𝑇_𝑏+(1−𝑝)𝑝^′𝑎₀(𝑜_𝜎2+𝑇_𝑠−DIFS)[𝑜_𝜎2+𝑇_𝑠−DIFS]+(1−𝑝)𝑝^′[1−𝑎₀(𝑜_𝜎2+𝑇_𝑠−DIFS)][𝑜_𝜎2+𝑇_𝑠]+(1−𝑝)(1−𝑝^′)𝑝^′′𝑎₀(𝜎₂+𝑜_𝜎1+𝑇_𝑠−DIFS)[𝜎₂+𝑜_𝜎1+𝑇_𝑠−DIFS]+(1−𝑝)(1−𝑝^′)𝑝^′′[1−𝑎₀(𝜎₂+𝑜_𝜎1+𝑇_𝑠−DIFS)][𝜎₂+𝑜_𝜎1+𝑇_𝑠]+(1−𝑝)(1−𝑝^′)(1−𝑝^′′ℎ)𝜎,_𝑖,0=𝑝𝑇_𝑐+(1−𝑝)𝑇_𝑠−(1−𝑝)𝑃_empty𝑎₀(𝑑_𝑖ℎ)DIFS,0≤𝑖≤𝑚,_𝑖,𝑘=ℎ_−1,2,0≤𝑖≤𝑚,1≤𝑘≤CW_𝑖. (4.9)

The probability 𝑃_empty can be determined based on the fact that, by PASTA theorem and Burke's theorem [20], the stationary distributions of the tagged station at an arbitrary time point and immediately after an arbitrary completion time point of successful transmission are identical. The probability that the tagged station is empty at an arbitrary time point is[𝑏_−2,0{1−𝑎₀(DIFS)}+𝑏_−1,5{1−𝑝𝑎₀(𝑇_𝑏−DIFS)−(1−𝑝)𝑝^′𝑎₀(𝑜_𝜎2+𝑇_𝑠−DIFS)−(1−𝑝)(1−𝑝^′)𝑝^′′𝑎₀(𝜎₂+𝑜_𝜎1+𝑇_𝑠−DIFS)−(1−𝑝)(1−𝑝^′)(1−𝑝^′′)𝑎₀×1(𝜎)}]𝜆[𝑏_−2,0ℎ_−2,0+∑⁵_𝑘=0𝑏_−1,𝑘ℎ_−1,𝑘+∑^𝑚_𝑖=0∑^CW𝑖_𝑘=0𝑏_𝑖,𝑘ℎ_𝑖,𝑘], (4.10) and the probability that the tagged station is empty immediately after an arbitrary completion time point of successful transmission at the tagged station is(1−𝑝)𝑃_empty∑^𝑚_𝑖=0𝑎₀(𝑑_𝑖)𝑏_𝑖,0+(1−𝑝)(1−𝑝^′)𝑎₀(𝑇_𝑠)𝑏_−1,0+(1−𝑝)(1−𝑝^′)(1−𝑝^′′)𝑎₀(𝑇_𝑠)𝑏_−1,1𝑏_0,0+(1−𝑝)(1−𝑝^′)𝑏_−1,0+(1−𝑝)(1−𝑝^′)(1−𝑝^′′)𝑏_−1,1. (4.11)Thus,𝑃_empty=Num.of𝑃_empty(1−𝑝)∑^𝑚_𝑖=0𝑎₀(𝑑_𝑖)𝑏_𝑖,0, (4.12)whereNum.of𝑃_empty=1𝜆[𝑏_−2,0ℎ_−2,0+∑⁵_𝑘=0𝑏_−1,𝑘ℎ_−1,𝑘+∑^𝑚_𝑖=0∑^CW𝑖_𝑘=0𝑏_𝑖,𝑘ℎ_𝑖,𝑘]×[𝑏_−2,0{1−𝑎₀(DIFS)}+𝑏_−1,5{1−𝑝𝑎₀(𝑇_𝑏−DIFS)−(1−𝑝)𝑝^′𝑎₀(𝑜_𝜎2+𝑇_𝑠−DIFS)−(1−𝑝)(1−𝑝^′)𝑝^′′𝑎₀(𝜎₂+𝑜_𝜎1+𝑇_𝑠−DIFS)−(1−𝑝)(1−𝑝^′)(1−𝑝^′′)𝑎₀(𝜎)}]×[𝑏_0,0+(1−𝑝)(1−𝑝^′)𝑏_−1,0+(1−𝑝)(1−𝑝^′)(1−𝑝^′′)𝑏_−1,1]−(1−𝑝)(1−𝑝^′)𝑎₀(𝑇_𝑠)𝑏_−1,0−(1−𝑝)(1−𝑝^′)(1−𝑝^′′)𝑎₀(𝑇_𝑠)𝑏_−1,1. (4.13)From (4.1)–(4.12), the stationary probability 𝑏_𝑖,𝑘 can be found by a numerical method.

The system throughput 𝑆, the fraction of time used for successful payload transmission, can be expressed as𝑆=Num.of𝑆Den.of𝑆, (4.14)withNum.of𝑆=[𝑃_tr𝑃_𝑠+(1−𝑃_tr)𝑃₁+(1−𝑃_tr)(1−𝑃₁)𝑃₂]𝐸[Payload],Den.of𝑆=𝜎+𝑃_tr[𝑃_𝑠𝑇_𝑠+(1−𝑃_𝑠)𝑇_𝑐]+(1−𝑃_tr)[𝑃₁(𝑜_𝜎2+𝑇_𝑠)+(1−𝑃₁)𝑃₂(𝜎₂+𝑜_𝜎1+𝑇_𝑠)], (4.15)where the meaning of 𝑃_tr, 𝑃_𝑠, 𝑃₁, and 𝑃₂ is as follows: 𝑃_tr is the probability that there is at least one transmission at the beginning of a slot time, with 𝑛 stations contending for the channel, each transmitting with probability 𝜏. Thus,𝑃_tr=1−(1−𝜏)^𝑛. (4.16)The probability 𝑃_𝑠 is the conditional probability that a packet transmission occurring on the channel at the beginning of a slot time is successful. This event corresponds to the case in which exactly one station transmits at that time. Thus,𝑃_𝑠=𝑛𝜏(1−𝜏)^𝑛−1𝑃_tr. (4.17)The probability 𝑃₁ is the conditional probability that a packet transmission during the first 𝜎₂ period of a slot time is successful, given that there are no transmission at the beginning of the slot time. This event corresponds to the case in which at least one station is in state (−1,0) at the beginning of the slot time, given that all the stations are in state (−1,0) or state (𝑖,𝑗) for 𝑗≠0 at the beginning of the slot time. Thus,𝑃₁=1−(1−𝜏^′)^𝑛. (4.18)The probability 𝑃₂ is the conditional probability that a packet transmission during the last 𝜎₁ period of a slot time is successful, given that there are no transmission at the beginning of the slot time and during the first 𝜎₂ period of the slot time. This event corresponds to the case in which at least one station is in state (−1,1) at the beginning of the slot time, given that all the stations are in state (𝑖,𝑗) for 𝑗≠0 at the beginning of the slot time. Thus,𝑃₂=1−(1−𝜏^′′)^𝑛. (4.19)

5. Numerical Examples

In this section, we evaluate the performance of IEEE 802.11 DCF under different maximum backoff stages and different length of packet payload conditions through simulation and analytical results. We have developed a C++ simulator modeling both the DCF protocol details in IEEE 802.11 and the backoff procedures of a specific number of independent transmitting stations. The simulation also takes into account real operations of each transmitting station. In our experiments, we use the parameter setting in Table 1 [4]. Since we assume that no hidden terminals exist, this section deals only with the basic access method without RTS/CTS. We let the packet arrivals to any station be a Poisson process with the same rate 𝜆 (packets/s).

Table 1 

System parameters.

Figure 4 shows the throughput of DCF with 𝑚=7 as the packet arrival rate 𝜆 varies. For given 𝑛, the throughout linearly increases as 𝜆 increases until the network is saturated. It drastically decreases just before a saturation point, and then maintains constant even though 𝜆 increases. Since traffic intensity increases as 𝑛 increases, the saturation point decreases. In addition, the throughput for large 𝑛 more sharply increases than that for small 𝑛. For all values of 𝑛, the maximum throughput is shown to be higher than the saturation throughput. The difference between the maximum throughput and the saturation throughput increases as 𝑛 increases. For 8184 bits of packet payload and 𝑛= 10, 20, 30, 40, and 50, the normalized maximum throughputs are about 0.78, 0.73, 0.70, 0.68, and 0.67, respectively, and the normalized saturation throughputs are about 0.76, 0.70, 0.66, 0.63, and 0.61, respectively. In case of the payload of 1024 bits, the normalized maximum throughputs are about 0.463, 0.447, 0.434, 0.425, and 0.418, respectively, and the normalized saturation throughputs are about 0.455, 0.429, 0.411, 0.396, and 0.385 for 𝑛= 10, 20, 30, 40, and 50, respectively.

(a)

(b)

(c)

(d)

Figure 4 

Throughput of DCF for IEEE 802.11 in case of 𝑚=7; (a) length of packet payload = 8184 bits, (b) 4096 bits, (c) 2048 bits, and (d) 1024 bits.

For 𝑚=5, the throughputs of DCF for IEEE 802.11 with different lengths of packet payload are shown in Figure 5. The general trends of throughputs with 𝑚=5 are similar to the case with 𝑚=7. In addition, the results show that the normalized maximum throughputs and the normalized saturation throughputs are nearly equal to those with 𝑚=7, because the maximum contention window size is enough to effectively resolve collisions under given parameters.

(a)

(b)

(c)

(d)

Figure 5 

Throughput of DCF for IEEE 802.11 in case of 𝑚=5; (a) length of packet payload = 8184 bits, (b) 4096 bits, (c) 2048 bits, and (d) 1024 bits.

6. Conclusions

We proposed a mathematical model to evaluate the performance of the IEEE 802.11 DCF protocol under unsaturation conditions. Even though the proposed model does not consider the post-backoff procedure, its results are shown to be very close to the simulation results. MAC throughput of DCF linearly increases as packet arrival rate increases until the network is saturated. When the network is saturated, MAC throughput becomes constant for various packet arrival rates.

Acknowledgment

This work was supported by the Ministry of Information and Communication (MIC), Korea, under the Information Technology Research Center (ITRC) support program supervised by the Institute of Information Technology Assessment (IITA).