4.5 Optimization of Uplink Bandwidth Request Transmission Mechanism
4.5.2 Analytical Derivation of Performance of Random Access
In this section, the performance of random access for BW-REQ transmission is analytically derived.
The TBEB algorithm has been extensively analyzed for Wireless LAN in [Bia00]. The key assumption of the analysis for random access is that, at each transmission attempt, re-gardless of the number of retransmissions suffered, each BW-REQ transmission collides
with constant and independent probabilityp. This assumption leads to more accurate re-sults as long as the numberK of users and the back-off windowWiget larger [Bia00]. The probabilitypis referred to as conditional collision probability, meaning the probability of a collision seen by a request transmitted.
The average delay measured in numbers of frames can be expressed as a function of the conditional collision probabilitypas follows. Since the back-off counter at thei-th stage is uniformly chosen between 0 andWi−1, and the number of frames the user has to wait before transmission is uniformly distributed between 0 andLi −1, the resulting average delay at stagei, denoted withd¯i, is given by
d¯i = 1 2iL
2iL−1
X
j=0
j = 2iL−1
2 . (4.56)
As the collision happens with the probability ofp, the average delay over all stages has a geometrical distribution and is calculated as
d¯= 1 + (1−p)
m
X
i=0
(pi
i
X
j=0
d¯j) + X∞ i=m+1
(pi
m
X
j=0
d¯j + (i−m) ¯dm)
!
. (4.57)
By substituting (4.56) into (4.57), the average delayd¯of the BW-REQ transmission using random access can be expressed as the function of the conditional collision probabilityp as
d¯= 1
2(1−p) +L(1−p−2mpm+1)
2(1−2p)(1−p) . (4.58) In the following, the analytical derivation of the conditional collision probability p is presented. Adopt the discrete time scalet, withtandt+ 1representing the beginning of two consecutive TOs. Letc(t)be the stochastic process representing the back-off counter, ands(t)be the stochastic processes representing the back-off stage of a given user. It has been shown in [Bia00] that the bi-dimensional process{s(t), c(t)} can be modeled with the discrete-time Markov chain depicted in Figure 4.13. With
P{i1, n1|i0, n0}=P{s(t+ 1) =i1, c(t+ 1) =n1|s(t) =i0, c(t) =n0}, (4.59) the Markov chain is described by the following transition probabilities,
P{i, n|i, n+ 1}= 1, i∈[0, m], n∈[0, Wi−2]; (4.60) P{0, n|i,0}= 1−p
W0 , i∈[0, m], n∈[0, Wi−1]; (4.61) P{i, n|i−1,0}= p
Wi
, i∈[1, m], n∈[0, Wi−1]; (4.62) P{m, n|m,0}= p
Wm
, n∈[0, Wi−1]. (4.63)
4.5 Optimization of Uplink Bandwidth Request Transmission Mechanism
0,1
0, 0 0, 2 1
1 11p 1
1 W0
1 1
1 W0
1 1 1 1 1
,1 i , 0
i i, 2 1 i W, i11
1, 0 i1
1 11p 1
p Wi
1 1
1
p Wi1
1 1 1 1 1
,1 m , 0
m m, 2 1 m W, m11
1 11p 1
p Wm
1 1
p Wm
11p
0,W011 p
Collision Successful Transmision Restart backoff process
Figure 4.13: Markov chain model of the TBEB algorithm [Bia00].
[Bia00]. (4.60) accounts for the fact that the back-off counterc(t)is decreased by 1 at the beginning of each TO. (4.61) describes the fact that following a successful transmission with probability(1−p), a new transmission starts with back-off stage0and the back-off counter is uniformly chosen from0toW0 −1. (4.61) describes that following a collision with probability p, the back-off stage is increased by 1 to i and the back-off counter is uniformly chosen from0toWi−1accordingly. Let
bi,n= lim
t→∞P{s(t) =i, c(t) =n}, i= 0, . . . , m, n= 0, . . . , Wi−1 (4.64) denote the stationary distribution of the Markov chain. It has been shown [Bia00] that all the valuesbi,ncan be expressed as functions of the valueb0,0and of the conditional collision probabilityp, i.e.
bi,n = Wi−n Wi
bi,0, i∈[0, m], n∈[1, Wi−1], (4.65) bi,0 =
pib0,0, i∈[1, m−1]
pm
1−pb0,0, i=m . (4.66)
By summarizing the probabilities of the states in which c(t)is equal to zero, the proba-bility of a user to transmit in a frame, denoted withptx, is given by
ptx=P{c(t) = 0}=
m
X
i=0
bi,0 = b0,0
1−p. (4.67)
So far, the Markov chain model for the TBEB algorithm proposed in [Bia00] has been reviewed. However, this model cannot be directly applied to the considered system which is a frame-based system, as explained in the following.
In the considered frame-based system, there are two situations in which the back-off process will not continue, and so additional idle states have to be introduced correspond-ingly.
Firstly, after a transmission attempt, the user does not immediately know whether the transmission is successful or not until the beginning of the next frame, and so it will wait till the next frame to continue the back-off process. That is, suppose the transmission attempt happens in then-th TO of the totalN, the back-off process of that user will wait forN −nTOs before continuing. Because the back-off counter is uniformly chosen from Wi values which is equal to multiples of N, the distribution of the number of TOs that the user will wait before the re-start of the back-off process is uniformly distributed over [0, K−1]. Thus, the author of this thesis suggests, in [VZLT05], introducing(N −1)idle states, depicted in rectangles, in the Markov chain, as shown in Figure 4.14.
4.5 Optimization of Uplink Bandwidth Request Transmission Mechanism
1 2 1 N11
1 1
1 N
1 1
Transmission attempt
1
Figure 4.14: Markov chain model of the idle states after transmission attempt in the frame-based TBEB algorithm.
Let x(t) denote the number of TOs to wait before the back-off process continues, the corresponding transition probabilities are
P{x(t+ 1) =n|x(t) =n+ 1}= 1, n= 1, . . . , N −2
P{x(t+ 1) =n|transmission attempt}= N1, n= 1, . . . , N −1. (4.68) With the probability of transmission attemptptx, the stationary distribution of the(N−1) idle states after a transmission attempt
xn = lim
t→∞P{x(t) =n}, n∈[1, N −1] (4.69) is derived from (4.68) as
xn = N −n
N ptx. (4.70)
Secondly, after a successful transmission, the back-off processing will not restart if there is no BW-REQ to be transmitted. It is assumed that the BW-REQ is generated or updated in the beginning of each frame and include all the packets arrived in the last frame. Since new packet arrives in each frame with probabilityλ, after a successful transmission, the probability of a new generated BW-REQ is equal toλ. Thus, when there is no BW-REQ after a successful transmission, whose probability is(1−λ), the back-off process will be stopped for the current frame, i.e. the nextN TOs. Therefore, N idle states, depicted in hexagons, are inserted after a successful transmission, as shown in Figure 4.15.
By letting y(t) denote the number of TOs to wait before the back-off process contin-ues after a successful transmission, the transition probability among the idle states in Fig-ure 4.15 is
P{y(t+ 1) =N −1|successful transmission}= 1−λ, P{y(t+ 1) =N −1|y(t) = 0}= 1−λ,
P{y(t+ 1) =n|y(t) =n+ 1}= 1, n ∈[0, N −2]
(4.71)
111
2 1
1 1 N1
1 1 1 1
0 1
1 Successful transmission
Figure 4.15: Markov chain model of the idle states after successful transmission in the frame-based TBEB algorithm.
With the probability of a successful transmission being equal toptx(1−p), the stationary distribution of theN idle states after a successful transmission
yn = lim
t→∞P{y(t) =n}, n ∈[0, N −1] (4.72) is given by
yn = y0, n ∈[1, N −1],
yN−1 = (1−λ)·(y0+ptx(1−p)). (4.73) From (4.73), it is obtained that
yn = 1−λ
λ ptx(1−p), n ∈[0, N −1]. (4.74) By substituting the expression of ptxin (4.67) into (4.70) and (4.74), the valuesxnand yncan be expressed as functions of the valueb0,0and of the conditional probabilityp, i.e.
xn = N −n
N ptx= N−n N
b0,0
1−p, n∈[1, N−1] (4.75) yn = 1−λ
λ b0,0, n∈[0, N −1]. (4.76) Recall that the values bi,n can be also expressed as functions of the value b0,0 and of the conditional probabilityp, as seen from (4.65) and (4.66). By imposing the normalization condition, i.e. the sum of the stationary probability of all states in the Markov chain is 1
1 =
m
X
i=0 Wi−1
X
j=0
bi,j +
N−1
X
n=1
xn+
N−1
X
n=0
yn, (4.77)
b0,0 can then be expressed as a function of conditional collision probabilityp, i.e.
b0,0 = 2(1−p)
(W0+ 1) +pW01−1(2p)−2pm + (N −1) + 2N1−λλ(1−p), (4.78)
4.5 Optimization of Uplink Bandwidth Request Transmission Mechanism
By substituting (4.78) into (4.67), the probability that a user transmits in a randomly chosen TO,ptx, is expressed as a function of the conditional probabilityp, i.e.
ptx= b0,0
1−p = 2
(W0+ 1) +pW01−1(2p)−2pm + (N −1) + 2N1−λλ(1−p). (4.79) Furthermore, as the conditional collision probabilitypis equal to the probability that at least one of the(N −1)other users transmit, it yields
p= 1−(1−ptx)N−1. (4.80)
From equations (4.79) and (4.80), the two unknowns, the conditional collision probabil-ity pand the transmission probability ptx, can be solved. Once the conditional collision probabilitypis known, the performance of the BW-REQ transmission in terms of the av-erage delay can be calculated according to (4.58).
In order to verify the presented analytical derivation of the performance for random ac-cess, the analytically computed results are compared with the results obtained by means of numerical simulation. The performance in terms of average delay is depicted as a function of the arrival rate of the BW-REQ,λ, in Figure 4.16, for two different sets of parameters adjustable in TBEB algorithm,Landm. The analytical results are represented in curves, and the simulated results represented with markers are values averaged over 5000 frames.
It can be seen in Figure 4.16(a) that, the analytical results meet the simulative ones quite well over different BW-REQ arrival rateλranging from 0 to 1.0. It is obvious that at high arrival rate of BW-REQ, polling is desired compared to random access. In Figure 4.16(b), the performance at the low arrival rate is reported in detail.