An example

We consider a GI/G/1 FCFS queue, whose inter-arrival times and service durations have Pareto (type II) distributions. We choose the Pareto distribution because it is often the benchmark indicator of heavy-tailed behaviour. The c.d.f.’s are:

A(x) = 1− 5 x+5 !10 ,x>0, andG(x) = 1− 1 x+1 !3 ,x>0,

for the inter-arrival time and service duration respectively. Denote by fA(x) and fB(x) the p.d.f.’s

of distributions ofAandGrespectively. Therefore we have

fA(x) = 2 5 x+5 !11 ,x> 0, and fB(x) = 3 (x+1)4,x>0,

3.1. Nearly perfect sampling of theGI/G/1 FCFSqueue with heavy tail distribution inputs35

tion of the service duration distribution has c.d.f. as

B0(x)= 1−

1

x+1

!2

,x>0, which is also Pareto.

In the first stage of transition, we couple the service durations of all possible chains. As shown in Figure 3.2, there are two curves. One is y = fB(x), and another is the shifted p.d.f. y = fB(x− Wn−). Uniformly draw a point (denoted by C0) under curve y = fB(x): sample E ∼G(·) andU ∼Unif(0,1), then let

D= fB(E)U. (3.6)

Thus (E,D) is the coordinate of pointC0. Draw a horizontal line (y = D) across pointC0. It

intersects with the area under curvey = fB(x). The width of the intersected line segment is d,

and d= H, which is the x-coordinate of the intersection of curvey= fB(x) and liney= D.

It is clear that fB(H)= D. Combining with equation 3.6, we have fB(E)U = fB(H) ⇒ H = (E+1)U

−1/4₋

1

⇒ d=(E+1)U−1/4−1.

Starting from pointC0draw pointsCi,i= 0,1, . . ., with interval of d. It is easy to see that there

is exactly one of such points under each of the shifted p.d.f.’s curve. For anyWn− ∈ (xi−1,xi),

wherexi is the x-coordinate of pointCi and we definex−1 = 0, the coupled service duration is Bn = xi−Wn−. ThereforeWn+= xi = E+ l_W n−−E d m d, as shown in equation (3.3).

In the second stage of transition, we couple the inter-arrival times of all possible chains. The procedure is quite similar to what we did above and we have

d0 =(E0+5)U0−1/11−5, whereE0∼ A(·) andU0 ∼Unif(0,1).

The differences are those: we turn over the p.d.f. curves horizontally, and the unfinished workload will be truncated to 0 whenW(n+1)−is updated to be negative.

0 x y Wn− E C0 C1 C2 C3 C4 C5 D H Wn+

Figure 3.2: Coupling the service durations in the first stage of transition with the Multishift Coupling method. inter-arrival time is An =Wn+−xi−1. SoW(n+1)− =(xi−1)+= −E0+jWn++E0 d0 k

d0+, as demonstrated by equation (3.4). This procedure is illustrated in Figure 3.3.

Because the coupled chains might not coalesce in each trial, we need to consider the bias introduced by ignoring this possibility. Let

1 = Pr(Not capturing the stationary chain with range (0,x0) at−T0);

2 = Pr(No coalescence happens in the first trial).

According to equation (3.5), we have

1 ≈

ρ

1−ρB¯0(x0). (3.7)

Let =10−10be the total variation distance, then it should be that

3.1. Nearly perfect sampling of theGI/G/1 FCFSqueue with heavy tail distribution inputs37 0 x y Wn+ −E' C₀ C₋₁ D' C₁ C₂ C₃ −H' W(n+1) −

Figure 3.3: Coupling the inter-arrival times in the second stage of transition with the Multishift Coupling method.

by considering a worse case that the coalescence happens in the third trial. Given we have captured the stationary chain at time−Tm,m = 0,1,with range (0,x0), then the probability of

not capturing it at time −Tm+1 with the same range is less than 1, because the unfinished

workload in this chain is positively correlated. This argument leads to the term of 31 in

inequality (3.8).

For the second or third trial, the probability of no coalescence occurring is less than 2.

Since these three trials are independent, we obtain the term of₂3in inequality (3.8). The value of 2 is estimated with the simulated data by using Chebyshev’s inequality [49, p. 78]. It is

a loose estimation, so if we want to control the probability of bias occurring less than with one trial, i.e. 1+2 < , thenT0will become quite large (with order of 1010). So we consider

the multiple trials situation to estimate the upper bound of the bias probability, as illustrated in Figure 3.4.

Since the system evacuates the unfinished workload by E(A) − E(B) = 0.0556 in each

transition, the expected number of transitions for Wn− to return the average level is around

x0/0.0556). By tuning the value ofx0andT0 to accommodate inequality (3.8), we have x0 =5.6×105andT0 =1.41×107.

Time

−3T0 −2T0 −T0 0

0

x0

Figure 3.4: A worse case of applying CFTP to the GI/G/1 FCFS queue which has Pareto inter-arrival time and service duration distributions. The coalescence occurs in the third trial.

So with equation (3.7), it follows1≈ 2.87×10−11.

After driving back the extreme large state (x0) to a normal size with around 10 million steps,

the remaining 4 million transitions are enough to ensure these chains with “close” distance to coalesce with high probability. Note that we only need to follow two chains which were started from states 0 andx0, due to the monotonicity of this coupling.

By repeating the single-trial simulations for 1,000 times, we obtain the numbers of steps till coalescenceXi,i= 1, . . . ,1000. It follows that

2 = Pr(X> T0)≤

Var(X) (T0−E(X))2

≈2.2867×10−4

31+₂3 = 9.81×10−11 < .

Because the coalescence is a large probability event, all these 1,000 simulations coalesced in the first trial. Table 3.1 demonstrates some statistical results of Xi,i = 1, . . . ,1000 and

sampled waiting times. With the sampled waiting times, the e.c.d.f. and its 95% confidence band are plotted in Figure 3.5. For t > 0, the standard deviation of Pr(W ≤ t) is estimated by ˆσ = √ ˆ p(1−p)ˆ √ N , where ˆp = PN i=11(Wi≤t)

N and N = 1000. Therefore the confidence band is

approximately ( ˆp−1.96 ˆσ,pˆ +1.96 ˆσ). Other point-wise confidence bands in this thesis are constructed in the same way.

3.1. Nearly perfect sampling of theGI/G/1 FCFSqueue with heavy tail distribution inputs39 0 20 40 60 80 100 0.0 0.2 0.4 0.6 0.8 1.0 Time

Figure 3.5: The e.c.d.f. from simulations of 1,000 independent draws of waiting times in the GI/G/1 FCFS queue using the CFTP algorithm. Inter-arrival time and service duration both have Pareto distributions. Shaded areas are point-wise 95% confidence bands.

Transitions for coalescence (X) Waiting times

Min. 9,885,290 0

Max. 10,318,845 335

Average 10,077,061 9.28

Variance 3,700,868,963 250

95% C.I. (10,073,290, 10,080,831) (8.29, 10.26)