Delay Analysis of IEEE 802.11 in Single-Hop Networks
Marcelo M. Carvalho J. J. Garcia-Luna-Aceves ∗
Department of Computer EngineeringUniversity of California, Santa Cruz
Santa Cruz, CA 95064 USA{carvalho, jj}@soe.ucsc.edu
Abstract
This paper presents an analytical model to compute theaverage service time and jitter experienced by a packetwhen transmitted in a saturated IEEE 802.11 ad hoc net-work. In contrast to traditional work in the literature, inwhich a distribution is usually fitted or assumed, we usea bottom-up approach and build the first two moments ofthe service time based on the IEEE 802.11 binary exponen-tial backoff algorithm and the events underneath its opera-tion. Our model is general enough to be applied to any typeof IEEE 802.11 wireless ad hoc network where the channelstate probabilities driving a node’s backoff operation areknown. We apply our model to saturated single-hop ad hocnetworks under ideal channel conditions. We validate ourmodel through extensive simulations and conduct a perfor-mance evaluation of a node’s average service time and jit-ter for both direct sequence and frequency-hopping spreadspectrum physical layers.
1. Introduction
During the past few years we have witnessed an ever-growing interest in wireless technologies and their appli-cation to portable devices. As the number of users of suchtechnologies has increased, the demand for real-time trafficand delay-sensitive applications has become more critical.Along the efforts to satisfy such needs, standards for wire-less local area networks (WLANs) have been proposed, andthe IEEE 802.11 medium access control (MAC) protocol[7] is the de facto standard and the most widely used nowa-days. In the IEEE 802.11, the main mechanism to accessthe medium is the distributed coordination function (DCF),which is a random access scheme based on the carrier sensemultiple access with collision avoidance (CSMA/CA). The
∗ This work was supported in part by CAPES/Brazil and by the U.S. AirForce under grant No. F49620-00-1-0330.
DCF provides two access schemes: the default, called ba-sic access mechanism, and an optional, four-way handshakescheme. The standard also defines the optional point coordi-nation function (PCF), which is a centralized MAC protocolthat uses a point coordinator to determine which node hasthe right to transmit. The PCF suppports collision free andtime bounded services. However, because the PCF cannotbe used in multihop or single-hop ad hoc networks, the DCFis the access network widely assumed, which implies vary-ing delays for all traffic. Curiously, the majority of the workon analyzing the performance of IEEE 802.11 DCF has con-centrated on its throughput [2, 3, 4, 11] and not much atten-tion has been given to analyzing its delay.
In this paper, we provide an analytical model to com-pute the average service time and jitter experienced by apacket when transmitted in a saturated IEEE 802.11 ad hocnetwork. In contrast to traditional work in the literature, inwhich a distribution is usually fitted or assumed [3, 5, 6, 8],we use a bottom-up approach and build the first two mo-ments of a node’s service time based on the IEEE 802.11binary exponential backoff algorithm and the events under-neath its operation. The strength of our model relies on thefact that it can be applied to many network scenarios. Thekey to its successful application is the knowledge of thechannel state probabilities driving a node’s backoff opera-tion. Here, we apply our model to saturated, single-hop adhoc networks with ideal channel conditions, operating un-der the four-way handshake mechanism of the DCF. For thiscase, the channel state probabilities we obtain are based onthe work by Bianchi [2], which provides a set of nonlinearequations that relates a packet’s collision probability withits transmission probability (in steady-state). We linearizeBianchi’s model and find simple equations to these quan-tities. The reason for our approximation is twofold: easeof computation and the need to better understand the im-pact of system parameters on channel and system probabil-ities (something that is not so clear under a nonlinear sys-tem of equations). We validate both our model and the lin-earized system through extensive simulations and conduct
Proceedings of the 11th IEEE International Conference on Network Protocols (ICNP’03)
a performance evaluation of a node’s average service timeand jitter for the direct sequence spread spectrum (DSSS)and frequency-hopping spread spectrum (FHSS) physicallayers under the same scenario. We investigate their perfor-mance as we vary such parameters as initial contention win-dow size, slot time size, packet size, and maximum backoffstage.
The rest of the paper is organized as follows. Section 2briefly reviews the DCF mechanism. Section 3 presents ouranalytical model. Following that, in Section 4, we validateour model through simulations. Section 5 presents a perfor-mance evaluation of both DSSS and FHSS physical layers.In Section 6 we present our conclusions.
2. The Distributed Coordination Function
The DCF describes two techniques for packet transmis-sion: the default, a two-way handshake scheme called ba-sic access mechanism, and an optional four-way handshakemechanism. In the basic access mechanism, a node moni-tors the channel to determine if another node is transmit-ting before initiating the transmission of a new packet. If thechannel is idle for an interval of time that exceeds the dis-tributed interframe space (DIFS), the packet is transmitted.Otherwise, the node monitors the channel until it is sensedidle for a DIFS interval, when it then generates a randombackoff interval for an additional deferral time before trans-mitting. This collision avoidance feature of the protocol in-tends to minimize collisions during contention among mul-tiple nodes. In addition, to avoid channel capture, a nodemust wait a random backoff time between two consecutivenew packet transmissions, even if the medium is sensed idlein the DIFS time.
DCF has a discrete-time backoff timer. The backoff timeris decremented only when the medium is idle and it isfrozen when the medium is sensed busy. After a busy pe-riod, the decrementing of the backoff timer resumes onlyafter the medium has been free longer than a DIFS period.A transmission takes place when the timer zeros out. Theslot size of the backoff timer is denoted by σ, and equalsthe time needed by any node to detect the transmission ofa packet by any other node. It is, therefore, dependent onthe physical layer and accounts for the propagation delay,the transmit-to-receive turn-around time, and the time tosignal the state of the channel to the MAC layer. At eachpacket transmission, the backoff time is uniformly chosenin the range (0,W − 1). The value W is called the con-tention window and depends on the number of failed trans-missions for a packet, i.e., for each packet queued for trans-mission, the contention window W takes an initial valueWmin that doubles after each unsuccessful packet transmis-sion, up to a maximum of Wmax (the values of Wmin andWmax are physical-layer specific). The contention window
remains at Wmax for the remaining attempts. This is the so-called exponential backoff scheme. In the sequel, each at-tempt to transmit a packet during the exponential backoffwill be referred to as a backoff stage. An ACK is transmit-ted by the destination node to signal the successful packetreception. The ACK is immediately transmitted at the endof the packet, after a period of time called short interframespace (SIFS). If the transmitting node does not receive theACK within a specified timeout, or if it detects the transmis-sion of a different packet on the channel, it reschedules thepacket transmission according to the given backoff rules.Figure 1(a) illustrates the basic access mechanism.
The four-way handshake mechanism involves the trans-mission of the request-to-send (RTS) and clear-to-send(CTS) control frames prior to the transmission of the actualdata frame. A successful exchange of RTS and CTS framesattempts to reserve the channel for the time duration neededto transfer the data frame under consideration. The rules forthe transmission of an RTS frame are the same as those fora data frame under the basic access scheme. After receivingan RTS frame, the receiver responds with a CTS frame af-ter a SIFS. After the successful exchange of RTS and CTSframes, the data frame can be sent by the transmitter afterwaiting for a SIFS interval. In case a CTS frame is not re-ceived within a predetermined time interval, the RTS is re-transmitted following the backoff rules as specified in thebasic access procedures described above. The frames RTSand CTS carry the information of the length of the packetto be transmitted. This information can be read by any lis-tening node, which is then able to update a network allo-cation vector (NAV) containing the information of the pe-riod of time in which the channel will remain busy. There-fore, when a node is hidden from either the transmitting orthe receiving node, by detecting just one frame among theRTS and CTS frames, it can suitably delay further trans-missions to try to avoid collisions. Figure 1(b) illustratesthe four-way handshake mechanism, which we simply callthe RTS/CTS mechanism.
3. Analytical Model
3.1. Service Time Characterization
As mentioned in Section 2, once a node goes to back-off, its backoff time counter decrements according to theperceived state of the channel. If the channel is sensedidle, the backoff time counter is decremented. Otherwise,it is frozen, staying in this state until the channel is sensedidle again for more than a DIFS, at which time its decre-menting operation is resumed. While the backoff timer isfrozen, only two mutually exclusive events can happen inthe channel: either a successful transmission takes placeor a packet collision occurs. Therefore, if we denote the
Proceedings of the 11th IEEE International Conference on Network Protocols (ICNP’03)
three possible events a node can sense during its backoffby Es = {successful transmission}, Ei = {idle channel},and Ec = {collision}, each of the time intervals betweentwo consecutive backoff counter decrements, which we call“backoff steps”, will contain one of these three mutually ex-clusive events. In other words, during a node’s backoff, thej-th “backoff step” will result in either a collision, a trans-mission, or the channel being sensed idle. We assume thatevents in successive backoff steps are independent, whichis a reasonable assumption if the WLAN is relatively largeand if the time a node spends on collision resolution is aboutthe same as the time the channel is sensed bus due to col-lisions by noncolliding nodes. In the DCF, a node finds outthat a collision has taken place if it does not receive theacknowledgment to its transmission after a certain timeout(the ACK Timeout in the basic access mechanism and theCTS Timeout in the RTS/CTS mechanism). In other words,if a collision happens in a backoff step, the colliding nodesare assumed to go through the collision resolution process inthis same backoff step and, therefore, can be ready for trans-mission in the following backoff step. This way, we avoiddependencies on the number of colliding nodes at previousbackoff steps.
Given the above considerations, let k denote the backoffstage at which a specific node is at a certain instant of time,and let nk be the number of backoff time slots randomlychosen at the k-th stage. Assuming that the events Ei, Es,and Ec have probabilities ps = P{Es}, pi = P{Ei}, andpc = P{Ec}, respectively, and given that these events areindependent and mutually exclusive at each backoff step,then the probability that in nk slots we have ri “idle slots”,rc “collision slots”, and rc “successful slots” is given by the
multinomial probability distribution
P{r |nk,p} =nk
ri! rc! rs!pri
i prcc prs
s , (1)
where r = [ri rc rs]T , p = [pi pc ps]T , pi + pc + ps = 1,and ri + rc + rs = nk. Let t = [σ tc ts]T , where σ is thetime used when the channel is sensed idle (i.e., one back-off slot), ts is the average time the channel is sensed busydue to a successful transmission, and tc is the average timethe channel is sensed busy due to a collision in the chan-nel. If we denote by T k
B(r;nk) the total backoff time spentat the k-th backoff stage when ri slots are idle, rc slotshave collisions, and rs slots have successful transmissionswithin the randomly chosen nk slots, then
T kB(r;nk) = rT t = σri + tcrc + tsrs. (2)
Note that the event E = { ri idle slots, rc collision slots,rs successful slots | nk} is the same as the event E′ ={backoff timer zeros out after riσ + rctc + rsts time slots| nk}. Therefore,
P{rT t |nk,p} = P{r |nk,p}. (3)
From the above results, the average time a node spendsat the k-th backoff stage when nk backoff steps are chosenis simply
tion of the randomly chosen value nk at the k-th back-off stage. We can finally compute the average backoff time
Tk
B at the k-th stage by averaging over nk as follows:
Tk
B =∑Wk−1
nk=0 Tk
B(nk)P{nk} (5)
=∑Wk−1
nk=0 nk(σpi + tcpc + tsps)/Wk = α(Wk − 1)/2,
where α = σpi + tcpc + tsps. This last result is quite intu-itive: it simply states that the average time a node spends atthe k-th backoff stage is nothing but the product of the aver-age number of backoff steps, (Wk−1)/2, times the averagebackoff step size α.
We are now able to consider the more general caseof the binary exponential backoff algorithm. Let Rk be a3 × k matrix whose columns are the k “counting events”ri, i = 1, 2, . . . , k of each backoff stage up to the k-thstage, i.e, Rk = [r1 r2 . . . rk]. We are interested in com-puting P{Rk |nk}, where nk = [n1 n2 . . . nk]T is a col-umn vector of the number of time slots chosen in each ofthe k stages. By our independence assumption, the eventsthat happen while a node is in its (k − 1)-th backoff stage
Proceedings of the 11th IEEE International Conference on Network Protocols (ICNP’03)
where the last equality expresses the independence, amongstages, on the randomly-chosen number of backoff steps.Given Rk and nk, the total backoff time can be computedas follows:
TB(Rk;nk) =∑k−1
i=1
(rT
i t + tc)
+ rTk t
=∑k
i=1rTi t + (k − 1)tc, (7)
where tc accounts for the time a node spends on collisionresolution (according to our previous remark). We can nowcompute the average time it takes to successfully transmit apacket after k backoff stages:
TB(nk) = E{TB(Rk;nk) |nk} =
=∑
Rk
[∑ki=1r
Ti t + (k − 1)tc
]P{Rk |nk}
=∑k
i=1TB(ni) + (k − 1)tc. (8)
By averaging over nk, and observing that the selectednumber of backoff steps at a specific backoff stage is inde-pendent of the selected number of backoff steps at previousstages, we have that
TB(k) = E{TB(nk)} =∑
nkTB(nk)P{nk}
=∑
nkTB(nk)
∏ki=1 P{ni}
=∑
nk
[∑ki=1 TB(ni) + (k − 1)tc
] ∏ki=1 P{ni}
=∑k
i=1 Ti
B + (k − 1)tc, (9)
where Ti
B is given by Eq.(5). This last result simply tellsus that the backoff time is a non-linear function of the dis-crete random variable K of the number of backoff stagesa node has to go through before transmitting a packet suc-cessfully. Consequently, the backoff time probability dis-tribution is the same as the probability distribution of thenumber of backoff stages K1, which in turn is directly re-lated to the probability that a packet is succesfully transmit-ted at the end of the k-th stage. Therefore, if we let qk bethe probability of success that a packet experiences whenit is transmitted at the end of the k-th backoff stage, andif we make the reasonable assumption that P{packet col-lides at the k-th stage | packet collided at the 1st, 2nd, . . . ,(k − 1)-th stages} = P{packet collides at the k-th stage}then,
P{K = k} =[∏k−1
i=1 (1 − qi)]qk. (10)
1 In one-to-one mappings of discrete random variables, if y = g(x)then P{y = y} = P{x = x}[10].
Note that if the probabilities qi are independent of thebackoff stage and constant, i.e., qi = q, ∀ i ∈ N, then wesimply have the geometric distribution
P{K = k} = (1 − q)k−1q. (11)
For simplicity, let us assume from now on that qi =q, ∀ i ∈ N. In fact, very accurate throughput results wereobtained by Bianchi [2] by assuming a constant and inde-pendent collision probability. Given that, we can now com-pute the first two moments of the backoff time TB(k). Letus start with the average backoff time TB . From Eq.(9), wehave
TB = E{TB(k)} =∑∞
k=1 TB(k)P{K = k} (12)
=∑∞
k=1
[(∑ki=1 T
i
B
)+ (k − 1)tc
](1 − q)k−1q
=∑∞
k=1
(∑ki=1
α2 Wi
)(1 − q)k−1q − α
2q + (1−q)q tc.
To compute the first term of Eq.(12), we first observe that
Wi ={
2i−1Wmin if 1 ≤ i ≤ m2mWmin if m < i
(13)
where m is the “maximum backoff stage”, i.e., the valuesuch that Wmax = 2mWmin. We can now compute the re-maining summation in Eq.(12) by splitting it into two termsas follows:
∑∞k=1
(∑ki=1
α2 Wi
)(1 − q)k−1q =
=∑m
k=1
(∑ki=1
α2 Wi
)(1 − q)k−1q+
+∑∞
k=m+1
(∑ki=1
α2 Wi
)(1 − q)k−1q = S1 + S2.
For S1 we have:
S1 =∑m
k=1
(∑ki=1
α2 Wi
)(1 − q)k−1q =
= αWmin2
{2q
[1−[2(1−q)]m
1−2(1−q)
]+ (1 − q)m − 1
}.
To find S2, we notice first that, for k = m + 1,
∑ki=1
αWi
2 = αWmin2
∑m+1i=1 2i−1 = αWmin
2
(2m+1 − 1
).
Hence, for k = m + 2,
∑ki=1
αWi
2 = αWmin2
[∑m+1i=1 2i−1 +
∑m+2i=m+2 2m
]= αWmin
2
[(2m+1 − 1
)+ 2m
].
In general, for k = m + γ,
∑ki=1
αWi
2 = αWmin2
[∑m+1i=1 2i−1 +
∑m+γi=m+2 2m
]= αWmin
2
[(2m+1 − 1
)+ (γ − 1) · 2m
].
Proceedings of the 11th IEEE International Conference on Network Protocols (ICNP’03)
Making the change of variable j = k − (m + 1) in S2, wehave:
S2 = αWmin2
∑∞j=0
[(2m+1 − 1
)+ j · 2m
](1 − q)j+mq
= αWmin2
[(2m+1 − 1
)(1 − q)m + 2m(1−q)m+1
q
].
By adding S1 to S2 we obtain that the average backoff timeequals
TB =α(Wminβ − 1)
2q+
(1 − q)q
tc, (14)
where
β =q − 2m(1 − q)m+1
1 − 2(1 − q). (15)
Therefore, the average time a packet spends in backoff issimply the average number of backoff stages it goes through(1/q) times the average time it spends in each backoff stage,added to the respective average time spent on collision res-olution. Note here that the term Wminβ works as an “ef-fective window size”, scaling the initial contention-windowsize according to the maximum backoff stage m and thesuccess probability q. In the specific case in which thecontention window is constant at every backoff stage, i.e.,
Ti
B = T ∗ = α(W ∗ − 1)/2, ∀k ∈ N, TB(k) is simplyTB(k) = kT ∗ + (k − 1)tc. In this case, the average back-off time reduces to
TB =α(W ∗ − 1)
2q+
(1 − q)q
tc (16)
If we make m = 0 in (15), i.e., if we fix the contention win-dow size to the initial contention-window size, we have
TB =α(Wmin − 1)
2q+
(1 − q)q
tc. (17)
In the same way, if the contention window size is con-stant at every stage k, the variance of the total backoff timeis given by
Var {TB(k)} = Var{∑k
i=1Ti
B + (k − 1)tc}
= Var {kT ∗ + (k − 1)tc} = (T ∗ + tc)2(1−q)
q2
=[
α(W∗−1)2 + tc
]2(1−q)
q2 . (18)
In the case of the binary exponential backoff algorithm,we need to apply the same techniques we applied before tosquared and cross-product terms. For conciseness, we omithere the intermediate steps and give the final expression ob-tained after some algebra:
and, if we make m = 0 we obtain Eq.(18).Given the backoff time characterization, the average ser-
vice time equals
T = TB + ts, (21)
where ts is the time to successfully transmit a packet. Be-cause ts is a constant,
Var{T (k)} = Var{TB(k)}. (22)
Note that the service time distribution is the same as that ofthe backoff time, which, in this case, is a non-linear func-tion of a geometric random variable with parameter q.
3.2. Channel Probabilities
The model we have just presented is applicable when-ever the channel state probabilities p = [pi pc ps]T driv-ing a node’s backoff operation are known. In this Section,we compute the values of p for a saturated, single-hop adhoc network under ideal channel conditions. For this pur-pose, we rely on the work by Bianchi [2], which providesa model to evaluate the saturation throughput of the IEEE802.11 MAC protocol under the hypothesis of ideal chan-nel conditions (i.e., no hidden terminals and capture). Fol-lowing Bianchi’s analysis, we also assume a fixed numberof nodes, with each node always having a packet availablefor transmission, i.e., the transmission queue of each nodeis assumed to be always nonempty. The key approximationof his model, which we adopt here too, is that each packet,at each transmission attempt, collides with constant and in-dependent probability p = 1−q regardless of the number ofretransmissions suffered2. This probability is called the con-ditional collision probability, meaning that this is the prob-ability of a collision experienced by a packet being trans-mitted on the channel. Bianchi modeled the stochastic pro-cess representing the backoff time counter for a given nodeas a bidimensional discrete-time Markov process. Accord-ing to his development, the probability τ that a node trans-mits in a randomly chosen slot time is [2]
τ =2(1 − 2p)
(1 − 2p)(Wmin + 1) + pWmin(1 − (2p)m), (23)
which is a function of the conditional collision probabilityp, still unknown. To find the value of p, it is sufficient to notethat the probability p that a transmitted packet faces a col-lision in the channel is the probability that at least one of
2 Note that the probability q is the same as the one we used in Section3.1.
Proceedings of the 11th IEEE International Conference on Network Protocols (ICNP’03)
the n − 1 remaining nodes transmit in a given time slot. Bythe independence assumption given above, each transmis-sion experiences the system in the same state, i.e., in steadystate. Each remaining node transmits a packet with proba-bility τ in steady state. Therefore,
p = 1 − (1 − τ)n−1. (24)
Equations (23) and (24) form a nonlinear system in thetwo unknowns τ and p that can be solved using numeri-cal techniques. In fact, Bianchi showed [2] that this systemhas a unique solution. To make things simpler, and to bet-ter understand the effect of different parameters on thesetwo probabilities, we will find an approximate solution tothis nonlinear system by linearizing both equations. For thispurpose, let γ = 1 − τ be the probability that a node doesnot transmit in a randomly chosen slot time, i.e.,
Given the continuity of both γ(p) and its derivatives3 inthe interval p ∈ (0, 1), the Taylor series expansion of γ(p)at p = 0 is given by
γ(p) =Wmin − 1Wmin + 1
+2Wmin
(Wmin + 1)2p + O(p2), (26)
where O(p2) accounts for the second and high order termsin the Taylor series expansion. Hence, a first order approxi-mation of γ(p) is simply
γ(p) =Wmin − 1Wmin + 1
+2Wmin
(Wmin + 1)2p, (27)
which, in terms of q = 1 − p becomes
γ(q) =−2Wmin
(Wmin + 1)2q +
W 2min + 2Wmin − 1(Wmin + 1)2
≈ −2Wmin
(Wmin + 1)2q + 1. (28)
Given that τ = 1 − γ, we have
τ(q) =2Wmin
(Wmin + 1)2q =
2Wmin
(Wmin + 1)2(1 − p). (29)
Figure 2 shows the comparison between the nonlinear re-lationship of (23) with the linear approximation of (29) forDSSS parameters (Wmin = 32 and m = 5). The error inthe approximation becomes more significant as the colli-sion probability grows. However, given the range at whichτ is varying, the error tends to be very small. In Section 5we evaluate the performance of our approximation.
3 Continuity with respect to the critical value p = 1/2 can be shown bysimply rewriting γ(p) in the same way as it was done for τ(p) in [2].
0 0.2 0.4 0.6 0.8 10
0.01
0.02
0.03
0.04
0.05
0.06
0.07
Collision Probability
Tra
nsm
issi
on P
roba
bilit
y
Figure 2. Transmission probability τ : comparisonof nonlinear relationship versus linear approxima-tion.
We can now substitute our approximation of τ(q) in theequation that relates the probability that no node is transmit-ting at any randomly chosen slot time, i.e., q = (1− τ)n−1.Because 2Wmin/(Wmin + 1)2 << 1 and 0 < q < 1, wehave that
q =[1 − 2Wmin
(Wmin + 1)2q
]n−1
≈ 1 − 2(n − 1)Wmin
(Wmin + 1)2q
≈ (Wmin + 1)2
(Wmin + 1)2 + 2(n − 1)Wmin,
which leads to the following approximation for p:
p =2Wmin(n − 1)
(Wmin + 1)2 + 2Wmin(n − 1). (30)
Equations (29) and (30) clearly show the decoupling wehave achieved by linearizing the original system of equa-tions. Figure 3(a) shows the conditional collision proba-bility p as a function of the number of nodes n and theminimum congestion window Wmin. As we can see, forthe current parameters of the IEEE 802.11 protocol, i.e.,Wmin = 16 (FSSS) and Wmin = 32 (DSSS), the colli-sion probability is more than 50% if the number of nodes inthe wireless LAN exceeds 20 nodes.
0 50 100 150 2000
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Number of Stations
Col
lisio
n P
roba
bilit
y
Wmin = 16Wmin = 32Wmin = 64Wmin = 128
0 20 40 60 80 1000
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Number of Stations
Pro
babi
lity
Idle ChannelSuccessful TransmissionCollision
(a) (b)
Figure 3. (a) Collision probability as a functionof the number of nodes. (b) Conditional channelprobabilities.
Proceedings of the 11th IEEE International Conference on Network Protocols (ICNP’03)
We can now turn to the problem of finding the condi-tional channel probabilities, represented here by the vectorp. For this purpose, let Ptr be the probability that there is atleast one transmission in the considered time slot. Becausewe are considering the events experienced by a node dur-ing its backoff period, only the remaining n − 1 nodes canbe contending for channel access. Therefore, because eachof the remaining n − 1 nodes transmits a packet with prob-ability τ at steady state, we have
Ptr = 1 − (1 − τ)n−1. (31)
The probability Psuc that a transmission occurring on thechannel is successful is given by the probability that exactlyone node transmits on the channel, conditioned on the factthat at least one node transmits, i.e.,
Psuc =
(n−1
1
)τ(1 − τ)n−2
Ptr=
(n − 1)τ(1 − τ)n−2
1 − (1 − τ)n−1. (32)
Therefore, the probability that a successful transmissionoccurs in a given time slot is ps = P{Es} = PtrPsuc. Ac-cordingly, pi = P{Ei} = 1 − Ptr and pc = P{Ec} =Ptr(1 − Psuc). Figure 3(b) shows these three probabilitiesas a function of n, the number of nodes. Finally, for the timeintervals ts and tc, we follow the definition given by Bianchi[2], where4
In this section we evaluate the accuracy of our model inpredicting the first two moments of a node’s service time ina single-hop IEEE 802.11 WLAN. For this purpose, we usethe simulator Ns-2 [9] to run simulations on network sizesvarying from 8 to 56 nodes (in steps of 8). All nodes trans-mit to some other node in the network according to the sameCBR source rate with fixed packet sizes of 1500 bytes (IPpacket). We pick a source rate high enough to saturate thenodes for each network size. Nodes are randomly placedin an area of 20 × 20 meters and have no mobility. Eachrun corresponds to 6 minutes of data traffic. We trace eachnode in the network and compute both the mean and vari-ance of its service time. We repeat the experiment for 20 dif-ferent seeds. We do that not just for statistical reasons, but
4 It is shown in [6] that, for correct floor acquisition to occur, CTS pack-ets have to be at least the same size as RTS packets plus the turnaroundtime plus twice the propagation delay, which does not happen in theIEEE 802.11 protocol. We will ignore this and consider that collisionsinvolve RTS packets only.
also because of the fairness problem inherent in the IEEE802.11 DCF. As already reported in the literature [1, 11], theavailable bandwidth is not equally shared among competingnodes under the IEEE 802.11 protocol. We noticed the samebehavior during our simulations in some of the randomly-chosen topologies, where some nodes were more successfulin acquiring the channel than others. Regarding the physi-cal layer, we use Direct-Sequence Spread Spectrum (DSSS)with a raw bit rate of 2Mbps. Table 4 summarizes the pa-rameters used for our simulations. FHSS standard-specificparamaters are listed for completeness (ACK Timeout andCTS Timeout are not specified in the standard). We com-
pute the average service time and jitter of each node in eachrun, and take the average over all nodes in the network. Werepeat this computation for all 20 seeds and report the re-sults averaged over the 20 seeds. Figure 4(a) shows the nu-merical results for the average service time for both sim-ulations and analytical models (linear and nonlinear). Aswe can see, our analytical model performs quite well, es-pecially in small to medium-size networks, providing uswith an upper bound on the average service time. Regard-ing the increasing discrepancy observed as the number ofnodes grows, we note two main reasons. First, in our analyt-ical model, a packet can backoff infinitely in time, whereasin simulations (as in the standard) retry counters help theMAC determine when it is no longer worth it to continueattempting to transmit a packet. Therefore, only packetsthat were not discarded had their service time consideredin the statistics. The second reason stems from our assump-tion that periods of collision experienced by colliding nodeshave the same duration as the periods in which the channelis sensed busy by noncolliding nodes. As mentioned before,this is not necessarily true, because the CTS timeout is usu-ally longer than the assumed tc, which lasts RTS + DIFS+ τ µsec for noncolliding nodes. Fortunately, this discrep-ancy is practically irrelevant if we note the high variance(jitter) of the service time as the number of nodes grows,and the fact that the average service time predicted by bothlinear and nonlinear models are within standard deviationof simulation results, as shown in Figure 4(a). Another im-
Proceedings of the 11th IEEE International Conference on Network Protocols (ICNP’03)
Figure 4. (a) Average service time: error barsshow standard deviation (jitter) in both simula-tions and analytical model. (c) Jitter magnitude.
portant result is shown in Figure 4(b), where we can seehow accurate our analytical model is in predicting the mag-nitude of the jitter experienced by each node in the network.The similarity is quite striking, with the jitter predicted bythe nonlinear model a little bit less than that in simulations.From Figures 4(a), and 4(b), we see that the linear model isa more conservative model, providing higher values for bothdelay and jitter. This is due to the fact that, for the same val-ues of n and Wmin, the probability of having transmissionsand collisions in the channel during a node’s backoff timeis usually higher for the linear model than for the nonlinearmodel. Consequently, the delay and jitter are also higher.
5. Performance Evaluation
This section addresses the impact of some of the IEEE802.11 parameters on the average service time and jitter forboth DSSS and FHSS physical layers, based on the modelwe developed in Section 3 for saturated networks. Unlessstated otherwise, the parameters used are the ones in Table4. First, we consider the impact of the initial contention-window size on the average service time and jitter. Fig-ures 5(a) and 5(c) show the results for the DSSS physicallayer and Figures 5(b) and 5(d) show the results for theFHSS physical layer. From the results, we see that, over-all, DSSS performs better than FHSS in both average ser-vice time and jitter. In particular, if we look at the perfor-mance for their real parameters (Wmin = 32 for DSSSand Wmin = 16 for FHSS), we see that FHSS averageservice time is, roughly speaking, twice the values of theDSSS physical layer, specially for large networks. DSSSand FHSS exhibit the same behavior in terms of jitter. Animportant observation to be made here is that, as far as de-lay and jitter in saturated networks is concerned, increas-ing the initial contention-window size improves the perfor-mance of the system in both physical layers. Figures 5(e),5(f), 5(g), and 5(d) show very clearly the impact of the ini-tial contention-window size on service time and jitter. The
results refer to window sizes of 8, 16, 32, 64, 128, 512, and1024. Both metrics drop dramatically as we increase the ini-tial contention-window size to values such as 512 or 1024.For small to medium-size networks (around 20 nodes) thejitter is very small and the average service time is practicallyconstant for window sizes higher than 128. For small val-ues of window sizes, DSSS still performs better than FHSS.Their performance becomes similar when window sizes arebigger than 128.
Figures 6(a), 6(b), 6(c), and 6(d) show the performanceof DSSS and FHSS physical layers for packet sizes of 32,64, 128, 512, and 1024 bytes (IP packets). We see againthat DSSS outperforms FHSS in both average service timeand jitter. From the graphs, we see that performance is notvery affected for medium-sized networks. However, the im-pact on system performance is more critical for large net-works, where a considerable increase in mean service timeand jitter is noticeable as packet size increases. This re-sult can be explained if we refer to Figure 3(b). In thisFigure, it is shown that, as the number of nodes grows,the probability of having a successful transmission in thechannel also grows, which directly affects the average slotsize α. Therefore, even though it is commonly stated thatthe RTS/CTS mechanism is throughput-effective when thepacket size increases [2], we are facing here a clear trade-offon delay/throughput performance as the number of nodesincreases. Figures 6(e) and 6(g) show the average servicetime and jitter as we vary the slot time size for the case ofthe DSSS physical layer. Figures 6(f) and 6(h) show the re-sults for the FHSS physical layer. Data packet size is fixedto 1024 bytes. From the graphs, we see that, even though wehave a big packet size, the slot time size has neglible impacton system performance for both DSSS and FHSS physicallayers. This result parallels the one reported by Bianchi [2],where throughput does not change much as we vary the slottime size. The fact is that, the amount of idle channel timestill remains marginal with respect to the time spent in trans-missions and collisions regardless of how much we increasethe slot size.
Figures 7(a), 7(b), 7(c), and 7(d) show the results quan-tifying the impact of the maximum backoff stage (param-eter m) on the service time for DSSS and FHSS physicallayers. The results show that, as far as service time and jit-ter are concerned, the binary exponential backoff algorithmcan be very harmful in large, saturated networks if the max-imum backoff stage is high. In both DSSS and FHSS we seethat, the fewer backoff stages, the better is the performance,specially for large networks. This fact suggests that, in sat-urated networks where nodes always have a packet ready tobe sent in the head of their queues, the binary exponentialbackoff algorithm seems to be inappropriate. In fact, nodeswill constantly have to backoff. However, according to ourresults, it is more effective to keep a constant, large con-
Proceedings of the 11th IEEE International Conference on Network Protocols (ICNP’03)
Figure 5. Average service time versus numberof nodes for different initial contention-windowsizes: (a) DSSS (b) FHSS. Jitter versus numberof nodes for different initial contention-windowsizes: (c) DSSS (d) FHSS. Average service timeversus initial contention-window size for differentnetwork sizes: (e) DSSS (f) FHSS. Jitter versus ini-tial contention-window size for different networksizes: (g) DSSS (h) FHSS.
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Figure 6. Average service time versus packet sizefor different network sizes: (a) DSSS (b) FHSS. Jit-ter versus packet size for different network sizes:(c) DSSS (d) FHSS. Average service time versusslot size for different network sizes: (e) DSSS (f)FHSS. Jitter versus slot size for different networksizes: (g) DSSS (h) FHSS.
Proceedings of the 11th IEEE International Conference on Network Protocols (ICNP’03)
Figure 7. Average service time versus maximumbackoff stage: (a) DSSS (b) FHSS. Jitter versusmaximum backoff stage: (c) DSSS (d) FHSS.
tention window size than to increase the size of the con-tention window exponentially. This way, nodes will be moreaggressive in acquiring the floor, providing lower delays.
6. Conclusions
In this paper, we presented an analytical model for com-putation of the average service time and jitter experiencedby a packet when transmitted in a saturated ad hoc networkin which the IEEE 802.11 DCF is used. Using a bottom-up approach, we built the first two moments of the servicetime based on the IEEE 802.11 binary exponential backoffalgorithm and the events underneath its operation. We pro-vided a general model that can be applied to many scenar-ios where the channel state probabilities that drive a node’sbackoff operation are known. Here, we applied our model tosaturated single-hop networks with ideal channel conditionsand we carried out a performance evaluation of a node’s av-erage service time and jitter for the DSSS and FHSS physi-cal layers. According to our results, as far as delay and jitterare concerned, DSSS performs better than FHSS. In addi-tion to this, we found that, in contrast to previous studies onthroughput in which the RTS/CTS mechanism was found tobe practically independent of the initial contention-windowsize and network size, these parameters have a major impacton system performance if delay is the metric in which weare interested. In this case, the higher the initial contention-window size, the smaller the average service time and jitterare, especially for large networks. On the other hand, if weconsider packet size, the opposite applies: the smaller the
packet, the smaller the average service time and jitter are.Regarding the slot time size, we found that it has neglibleimpact on delay performance for both DSSS and FHSS. Fi-nally, for the maximum backoff stage, the binary exponen-tial backoff algorithm was found to be harmful if both themaximum backoff stage and the number of nodes in the net-work are large. As far as delay in saturated IEEE 802.11networks is concerned, the binary exponential backoff al-gorithm seems to be inapropriate, and a large and constantcontention window size was showed to be more efficient,with packet sizes being selected according to the networksize.
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[2] G. Bianchi. Performance analysis of the IEEE 802.11 dis-tributed coordination function. IEEE Journal on SelectedAreas in Communications, 18(3):535–547, March 2000.
[3] F. Cali, M. Conti, and E. Gregori. Dynamic tuning of theIEEE 802.11 protocol to achieve a theoretical throughputlimit. IEEE/ACM Transactions on Networking, 8(6):785–799, Dec 2000.
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[6] C. L. Fullmer and J. J. Garcia-Luna-Aceves. Solutions to hid-den terminal problems in wireless networks. In Proc. ACMSIGCOMM 97, Cannes, France, September 1997.
[7] IEEE Standard for Wireless LAN Medium Access Control(MAC) and Physical Layer (PHY) Specifications, Nov 1997.P802.11.
[8] L. Kleinrock and F. A. Tobagi. Packet switching in radiochannels: Part I - carrier sense multiple-access modes andtheir throughput-delay characteristics. IEEE Transactions onCommunications, COM-23(12):1400–1416, 1975.
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Proceedings of the 11th IEEE International Conference on Network Protocols (ICNP’03)
