# Performance Analysis of IEEE 802.11 DCF under Nonsaturation Condition

Authors: Yutae Lee; Min Young Chung; Tae-Jin Lee
Journal: Mathematical Problems in Engineering
 (2008)

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## 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.


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## Body

## 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,CWmin−1],
where CWmin 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 CWmax 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 Figure1).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 pointt,
we define a stochastic process s(t) as follows:(3.1)s(t)={−2iftisanembeddedtype1;−1iftisanembeddedpointoftype2ortype3atwhichthebackoffcounterofthetaggedstationisnotactivatedyet;iiftisanembeddedpointatwhichthebackoffstageisifor0≤i≤m,wheremisthemaximumbackoffstage.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)
whens(t)=−2, b(t)=0;(2)
whens(t)=−1,(3.2)b(t)={0if2σ≤tp<DIFS,1ifDIFS−σ≤tp<2σ,2ifσ≤tp<DIFS−σ,3ifDIFS−2σ≤tp<σ,4if0<tp<DIFS−2σ,5iftp=0,where tp  denotes the time period elapsed since the
first packet arrived at the tagged station which was empty;(3)
when0≤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 pointt,
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≤CWi,
is the possible backoff counter value at the backoff stage i,
and CWi 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 tp after the first packet arrivals is 2σ≤tp<DIFS with probability α1, DIFS−σ≤tp<2σ with probability α2, σ≤tp<DIFS−σ with probability α3, DIFS−2σ≤tp<σ with probability α4,
and 0<tp<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 have(3.3)α1=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.(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 have(3.4)β1=1−e−λσ1,β2=e−λσ1−e−λσ.(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 have(3.5)a0(l)=e−λl.For analytical simplicity, the
probability a0(l)  is approximated as a0(E[l]) in this section.(4)
ol: 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 ol,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 ol,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,(3.6)ol¯≡E[ol]=∑k=1∞lk+1(λl)ke−λlk!(1−e−λl)=1−(1+λl)e−λlλ(1−e−λl).(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)
di: 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)
Pempty: 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(di)=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 asn.
Let li 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 Tb+σ,
where Tb denotes the required time for the freezing
time due to the busy channel. The time period Tb is obtained as(3.7)Tb=(n−1)τ(1−τ)n−2pTs+[1−(n−1)τ(1−τ)n−2p]Tc,where Ts denotes the required time for the successful
transmission of a packet and Tc 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 oσ2¯+Ts+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+oσ1¯+Ts+σ.
Thus, the mean li¯≡E[li] of li can be obtained as(3.8)li¯=CWi2×[p(Tb+σ)+(1−p)p′¯(oσ2¯+Ts+σ)+(1−p)(1−p′¯)p′′¯(σ2+oσ1¯+Ts+σ)+(1−p)(1−p′¯)(1−p′′¯)σ]=CWi2×[σ+pTb+(1−p)p′¯(oσ2¯+Ts)+(1−p)(1−p′¯)p′′¯(σ2+oσ1¯+Ts)].Using li¯,
the mean di¯ of di can be obtained as(3.9)di¯=Ts−DIFS+iTc+∑k=0ilk¯for 0≤i<m;
assuming no retransmission limit, we obtain(3.10)dm¯=Ts−DIFS+(m+p1−p)Tc+∑k=0m−1lk¯+lm¯1−p.


## 4. Mathematical Analysis

Letbi,j be the stationary probability of the described
Markov chain. The stationary probabilities satisfy the following balance
equations:(4.1)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;bi,0=pib0,0,1≤i<m;bm,0=pm1−pb0,0;bi,k=CWi+1−kCWi+1bi,0,0≤i≤m,0≤k≤CWi.From state transition diagram,
the expression for p0,0 is given by(4.2)b0,0={p+(1−p)p′+(1−p)(1−p′)(1−a0(Ts))}b−1,0+{p+(1−p)p′¯+(1−p)(1−p′¯)p′′+(1−p)(1−p′¯)(1−p′′)(1−a0(Ts))}b−1,1+{p+(1−p)p′¯+(1−p)(1−p′¯)p′′¯}(b−1,2+b−1,3+b−1,4)+{p(1−a0(Tb−DIFS))+(1−p)p′¯(1−a0(oσ2+Ts−DIFS))+(1−p)(1−p′¯)p′′¯(1−a0(σ2+oσ1+Ts−DIFS))}b−1,5+∑i=0m(1−p)(1−Pemptya0(di))pi,0={p+(1−p)p′+(1−p)(1−p′)(1−a0(Ts))}b−1,0+{p+(1−p)p′¯+(1−p)(1−p′¯)p′′+(1−p)(1−p′¯)(1−p′′)(1−a0(Ts))}b−1,1+{p+(1−p)p′¯+(1−p)(1−p′¯)p′′¯}(b−1,2+b−1,3+b−1,4)+{p(1−a0(Tb−DIFS))+(1−p)p′¯(1−a0(oσ2+Ts−DIFS))+(1−p)(1−p′¯)p′′¯(1−a0(σ2+oσ1+Ts−DIFS))}b−1,5+b0,0−(1−p)Pempty[∑i=0m−1a0(di)pi+pm1−pa0(dm)]b0,0.Thus,(4.3)b0,0=[{p+(1−p)p′+(1−p)(1−p′)(1−a0(Ts))}b−1,0+{p+(1−p)p′¯+(1−p)(1−p′¯)p′′+(1−p)(1−p′¯)(1−p′′)(1−a0(Ts))}b−1,1+{p+(1−p)p′¯+(1−p)(1−p′¯)p′′¯}(b−1,2+b−1,3+b−1,4)+{p(1−a0(Tb−DIFS))+(1−p)p′¯(1−a0(oσ2+Ts−DIFS))+(1−p)(1−p′¯)p′′¯(1−a0(σ2+oσ1+Ts−DIFS))}b−1,5]×1(1−p)Pempty[∑i=0m−1a0(di)pi+(pm/(1−p))a0(dm)].Hence, each of the stationary
probabilities can be expressed in terms of b−2,0,
which can be obtained from the normalization condition(4.4)b−2,0+∑k=05b−1,k+∑i=0m∑k=0CWibi,k=1.Each ofτ, 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 ∑i=0mbi,0,
the probability τ is given by(4.5)τ=∑i=0mbi,01−b−2,0.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 by(4.6)τ′=b−1,0(1−b−2,0)(1−τ).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 by(4.7)τ′′=b−1,1(1−b−2,0)(1−τ)(1−τ′).Having obtainedτ, τ′,
and τ′′,
the probabilities p, p′, p′¯, p′′,
and p′′¯ can be determined as follows:(4.8)p=1−(1−τ)n−1,p′=∑i=1n−1ii+1(n−1i)(τ′)i(1−τ′)n−i−1,p′¯=1−(1−τ′)n−1,p′′=∑i=1n−1ii+1(n−1i)(τ′′)i(1−τ′′)n−i−1,p′′¯=1−(1−τ′′)n−1.Lethi,j denote the mean sojourn time at state (i,j).
Then,(4.9)h−2,0=DIFS,h−1,0=pTb+(1−p)p′(oσ2+Ts)+(1−p)(1−p′)a0(Ts)(oσ2+Ts−DIFS)+(1−p)(1−p′)(1−a0(Ts))(oσ2+Ts),h−1,1=pTb+(1−p)p′¯(oσ2+Ts)+(1−p)(1−p′¯)p′′(σ2+oσ1+Ts)+(1−p)(1−p′¯)(1−p′′)a0(Ts)(σ2+oσ1+Ts−DIFS)+(1−p)(1−p′¯)(1−p′′)(1−a0(Ts))(σ2+oσ1+Ts),h−1,2=pTb+(1−p)p′¯(oσ2+Ts)+(1−p)(1−p′¯)p′′¯(σ2+oσ1+Ts)+(1−p)(1−p′¯)(1−p′′¯)σ,h−1,k=h−1,2,k=3,4,h−1,5=pa0(Tb−DIFS)[Tb−DIFS]+p[1−a0(Tb−DIFS)]Tb+(1−p)p′¯a0(oσ2+Ts−DIFS)[oσ2+Ts−DIFS]+(1−p)p′¯[1−a0(oσ2+Ts−DIFS)][oσ2+Ts]+(1−p)(1−p′¯)p′′¯a0(σ2+oσ1+Ts−DIFS)[σ2+oσ1+Ts−DIFS]+(1−p)(1−p′¯)p′′¯[1−a0(σ2+oσ1+Ts−DIFS)][σ2+oσ1+Ts]+(1−p)(1−p′¯)(1−p′′¯)σ,hi,0=pTc+(1−p)Ts−(1−p)Pemptya0(di)DIFS,0≤i≤m,hi,k=h−1,2,0≤i≤m,1≤k≤CWi.The probabilityPempty 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(4.10)[b−2,0{1−a0(DIFS)}+b−1,5{1−pa0(Tb−DIFS)−(1−p)p′¯a0(oσ2+Ts−DIFS)−(1−p)(1−p′¯)p′′¯a0(σ2+oσ1+Ts−DIFS)−(1−p)(1−p′¯)(1−p′′¯)a0(σ)}]×1λ[b−2,0h−2,0+∑k=05b−1,kh−1,k+∑i=0m∑k=0CWibi,khi,k], and the probability that the
tagged station is empty immediately after an arbitrary completion time point of
successful transmission at the tagged station is(4.11)(1−p)Pempty∑i=0ma0(di)bi,0+(1−p)(1−p′)a0(Ts)b−1,0+(1−p)(1−p′¯)(1−p′′)a0(Ts)b−1,1b0,0+(1−p)(1−p′)b−1,0+(1−p)(1−p′¯)(1−p′′)b−1,1.Thus,(4.12)Pempty=Num.ofPempty(1−p)∑i=0ma0(di)bi,0,where(4.13)Num.ofPempty=1λ[b−2,0h−2,0+∑k=05b−1,kh−1,k+∑i=0m∑k=0CWibi,khi,k]×[b−2,0{1−a0(DIFS)}+b−1,5{1−pa0(Tb−DIFS)−(1−p)p′¯a0(oσ2+Ts−DIFS)−(1−p)(1−p′¯)p′′¯a0(σ2+oσ1+Ts−DIFS)−(1−p)(1−p′¯)(1−p′′¯)a0(σ)}]×[b0,0+(1−p)(1−p′)b−1,0+(1−p)(1−p′¯)(1−p′′)b−1,1]−(1−p)(1−p′)a0(Ts)b−1,0−(1−p)(1−p′¯)(1−p′′)a0(Ts)b−1,1.From (4.1)–(4.12), the stationary
probability bi,k can be found by a numerical method.The system throughputS,
the fraction of time used for successful payload transmission, can be expressed
as(4.14)S=Num.ofSDen.ofS,with(4.15)Num.ofS=[PtrPs+(1−Ptr)P1+(1−Ptr)(1−P1)P2]E[Payload],Den.ofS=σ+Ptr[PsTs+(1−Ps)Tc]+(1−Ptr)[P1(oσ2+Ts)+(1−P1)P2(σ2+oσ1+Ts)],where the meaning of Ptr, Ps, P1,
and P2 is as follows: Ptr 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,(4.16)Ptr=1−(1−τ)n.The probability Ps 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,(4.17)Ps=nτ(1−τ)n−1Ptr.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,(4.18)P1=1−(1−τ′)n.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,(4.19)P2=1−(1−τ′′)n.


## 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 Table1 [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.
MAC header272 bitsPHY header128 bitsACK length112 bits + PHY headerChannel bit rate1 Mbit/sPropagation delay1μsSlot time50μsSIFS28μsDIFS128μsInitial contention window CW31Maximum contention window CW1023Maximum backoff stagem5 or 7Figure4 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.Throughput of DCF for IEEE 802.11 in case ofm=7;
(a)  length of packet payload = 8184 bits, (b) 4096 bits, (c) 2048 bits, and (d) 1024 bits.
(a)(b)(c)(d)Form=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.Throughput of DCF for IEEE 802.11 in case ofm=5;
(a) length of packet payload = 8184 bits, (b) 4096 bits, (c) 2048 bits, and (d) 1024 bits.
(a)(b)(c)(d)


## 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.



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*Source: 574197-2008-12-31.xml*