As before, the state of the GlMll queue consist of two parts, a continuous and a discrete part, since the state is given by the number of customers in the system and the time since the l
Trang 1Chapter 7
I N Chapters 5 and 6 we have addressed queues with generally distributed service time distributions, but still with Poisson arrivals In this chapter we focus on queues with more general interarrival time distributions In Section 7.1 we address the GlMll queue, the important “counterpart” of the MIGil q ueue Then, in Section 7.2, we present an exact result for the GIG11 q ueue Since this result is more of theoretical than of any practical interest, we conclude in Section 7.3 with a well-known approximate result for the GIG11 queue
It should be noted that most of the exact results presented in this chapter are less easy to apply in practical performance evaluation For a particular subclass of GIGI 1 queueing models, namely those where the interarrival and service times are of phase-type, easy-applicable computational techniques have been developed, known as matrix-geometric techniques These techniques will be studied in Chapter 8
For the analysis of the G IMI 1 queue, one encounters similar problems as for the analysis of the M/G/l q ueue As before, the state of the GlMll queue consist of two parts, a continuous and a discrete part, since the state is given by the number of customers in the system and the time since the last arrival Unfortunately, the intuitively appealing method of moments followed for the MI G I 1 q ueue, based on average values, the PASTA property and knowledge about the residual service time cannot be used in this case because the PASTA property does not hold Instead, we will simply state a number of important results and discuss
Performance of Computer Communication Systems: A Model-Based Approach.
Boudewijn R Haverkort Copyright © 1998 John Wiley & Sons Ltd ISBNs: 0-471-97228-2 (Hardback); 0-470-84192-3 (Electronic)
Trang 2134 7 GIMIl-FCFS and GIGIl-FCFS queueing models
their meaning in detail
We first have to define some notation The interarrival time distribution is denoted
FA (t), and has as first moment E[A] = l/X and as second moment E[A2] The service time distribution F,(t) = 1 - e- pt, t 2 0, E[S] = l/p, E[S2] = 2/p2 and Cg = 1
As we will see below, we need knowledge about the full interarrival time distribution and density; the latter is denoted f~(t) and has Laplace transform f;(s):
The Laplace transforms of many interarrival time densities are easy to derive; some of them are listed in Appendix B
For the G]M]l q ueue, an embedded Markov chain approach can be employed, similar to the one discussed in Section 5.3 for the M]G]l q ueue A general two-dimensional Markov chain can then be defined: ((N(t), V(t)), t 2 0), w h ere N(t) E IV denotes the number of customers in the queueing station and V(t) E R denotes the time since the last arrival; we have to keep track of this time since the interarrival times are not memoryless anymore Suitable embedding moments are now arrival instances, since then V(t) = 0 Taking this further will reveal that the probability that an arriving job finds i jobs in the queueing system is of the form
with 0 < 0 < 1 Surprisingly, this is a geometric distribution, just as in the MIMI1 case Notice, however, that the base of this geometric distribution is not p but 0, which is defined
as the probability that the system is perceived not empty by an arriving customer This agrees with the fact that ~0 = 1 - 0, and thus we also have that 0 = 1 - ro = ‘& ri Notice that in general 0 # p; only in case of Poisson arrivals are these two quantities the same, thus reflecting the PASTA property As a result of this, for general arrival processes, the probability that an arriving customer finds the queue non-empty differs from p The long-term probability that the queue is not empty, however, remains equal to p
Knowing the probabilities ri, we can compute the expected waiting time for a customer
as
co
by noting that for a customer arriving at a station with i customers in it, which happens with probability ri, i services of average length E[S] have to be performed before the arriving customer is being served Notice that we use the memoryless property of the
Trang 37.1 The GlMll queue 135 service time distribution here Continuing our computation, we find:
E[W] = 2 r&?qS] = F(l - a)a%E[S] = aE[S](l - 0) 2 iai-r
= OE[S](l - 0) F $(Oi) = OE[S](l - O)$ (gOi)
i-0
This is again a remarkable result since it has exactly the same form as the M]M] 1 result for the expected waiting time, however, with p replaced by 0 Hence, we can view the G]M] 1 queue as an M ] M 11 queue in which p is replaced by 0 through some transformation Without proof we now state that o can be derived from the following nonlinear equation involving the Laplace transform of the interarrival time density:
This fixed-point equation can be understood as the transformation of p in 0 Unfortunately, (7.5) can most often not be solved explicitly We therefore usually have to employ the following (straightforward) fixed-point iteration Pick a first guess for CT, denoted a(‘) It can be proven that 0 < 0 < 1 as long as p = XE[S] < 1, so a first guess could be o(l) = p Then, we compute ac2) = ji(p(l - &))), and so on, until we find a a@) such that (7.5)
is sufficiently well satisfied Notice that cr = 1 is always a solution of (7.5)) however, this solution is not valid as it would not result in a proper density fi
Example 7.1 Poisson arrivals
In case we have Poisson arrivals, fA(t) = Xemxt, and the above results should
results we already know for the MIMI 1 queue We find as Laplace transform
so that (7.5) becomes:
x
c7 = /Q(l - a) + x * (0 - l)(/Qa - A) = 0
reduce to
(7.6)
(7.7) The solution 0 = 1 is not valid; it would not result in a proper density for ri, Therefore, the result is 0 = X/p, which equals our expectation since for the MIMI 1 queue, the queue length distribution at arrival instances equals the steady-state queue length distribution
cl
Trang 4136 7 GlMll-FCFS and GlGll-FCFS queueing models
Example 7.2 The DIM11 queue
For the DIM]1 q ueue, we have FA(t) = l(t 2 l/X), that is, arrivals take place exactly every l/X time units The density f~(t) is a Dirac impulse at t = l/X (see Appendix B)
As Laplace transform we find fi(s) = e-8/X, so that we have to obtain 0 from
This nonlinear equation cannot be solved explicitly, so that we have to resort to the fixed- point iteration scheme
As a numerical example, consider the case where E[S] = 1.0 and where X increases from 0.05 to 0.95, i.e.,p increases from 5% to 95% In Table 7.1 we compare the results for the D(M(1 queue with those of the MIMI1 q ueue As performance measure of interest,
we have chosen the average waiting time As can be observed, the deterministic arrivals have a positive effect on the performance since E[W] is smaller in the deterministic arrivals case
Also observe that 0 approaches p as p increases As CT has taken over the role of p
in the expression for the mean waiting time, we see that for small utilisations (when CJ is much smaller than p) the waiting time is much lower than in the corresponding MIMI1 case For larger utilisations the effect of having deterministic arrivals instead of Poisson arrivals becomes relatively less important
To conclude this example, we can state that with respect to man waiting times the DIM] 1 queue can be seen as an MIMI 1 queue with reduced utilisation 0 where 0 follows
Since we have knowledge about the number of customers seen by an arriving customer,
we can express the waiting time distribution for such an arriving customer as well If
n customers are present upon its arrival, there are n services to be performed, before the arriving customer is taken into service Since all the services are of exponentially distributed length, the arriving customer perceives an Erlang-n waiting time distribution (denoted
FE, (t)) with probability r, This is similar to what we have already seen in Chapter 4, where we computed the response time distribution for the M(M(1 queue Summing over all possible numbers of customers present at arrivals, and weighting with the appropriate occurrence probabilities, we find
Pr{W 5 t} = E r,F~,(t) = g(l - c+PFEn(t)
n=O n=O
= (1 - 0) g on (I- epPtz $I.!)
i=o *
Trang 57.1 The GlMll queue 137
0.05 2.06 x lo-’ 2.06 x lo-’ 5.26 x 1O-2 0.15 1.28 x 1O-3 1.29 x 1O-3 1.77 x 10-l 0.25 1.98 x 1O-2 2.02 x 1O-2 3.33 x 10-l 0.35 7.02 x 1O-2 7.55 x 1O-2 5.39 x 10-l 0.45 1.52 x 10-l 1.79 x 10-l 8.18 x 10-i 0.55 2.61 x 10-l 3.83 x 10-l 1.22 0.65 3.93 x 10-l 6.48 x 10-l 1.86 0.75 5.46 x 10-l 1.20 3.00 0.85 7.16 x 10-l 2.52 5.67 0.95 9.01 x 10-l 9.17 1.90 x lOi
Table 7.1: Comparing the expected waiting time E[W] of a DIM]1 queue with an MIMI1 queue
= 1 - (1 - ++ E L$z 5 fy
i=o ’ n=i+l
i #+l
= 1- (l-+-+$-G
i=O * DC) (pcq
= 1 - ae-@ C 7
i=O ’
= 1 - ae-p(l-a)t, t 2 0
(7.9)
In a similar way, we find for the response time distribution:
Pr{R 5 t} = 1 - e-h(l-a)t, t 2 0 (7.10) Both these results are again similar to those obtained previously for the MIMI1 queue (see Section 4.4); the only difference is that 0 takes over the role of p
Prom the waiting time density fw(t) = a,~(1 - a)e-p(l-a)t, we can also derive the average waiting time as we have seen before
Example 7.3 The Hypo-2IM/l queue
In the Hypo-2]M]l q ueue services take an exponentially distributed time (here, we take
Trang 6138 7 GIMIl-FCFS and G 1 G ( l-FCFS queueing models
E[S] = l/p = 1) Th e interarrival times can be considered to consist of two exponential phases; only after an exponentially distributed time with rate X1 and an exponentially distributed time with rate X2 an arrival takes place (here we assume that Xr = 2 and X2 = 1) Thus, the mean interarrival time E[A] = l/2 + l/l = 1.5 and the utilisation
p = 2/3 From A ppendix A we know that
fA(t) = $$- (emXzt - eexlt), t 2 0
1 2
For the Laplace transform we find (with X1 = 2 and X2 = 1):
(7.11)
h x2
fxs) = (x,(s)2 + s) - (s + 2;(s + 1)’
The fixed-point equation becomes:
(l-0+2)2(1-0+1) = u + u3 - 5a2 + 60 - 2 = 0
(7.12)
(7.13) This third-order polynomial can be solved easily by noting that 0 = 1 must be a solution
to it (check this!) so that we can divide it by (0 - 1) yielding
This quadratic equation has solutions 0 = 2 f & Since we are looking for a solution
in the range (0, l), the only valid solution we find is CY = 2 - fi z 0.586 Note that D<P z 0.667 Thus, we find E[W] = (2 - fi)/(fi - 1) = 1.42 The waiting time distribution has the form
Fw(t) = 1 - (2 - JZ),-(J2P1)t, t 2 0, (7.15) and is depicted in Figure 7.1 Notice the ‘jump” at t = 0; it corresponds to the fact that there is a non-zero probability of not having to wait at all cl
Example 7.4 Waiting time distribution in the EklMll queue
We will now study the influence of the variance of the arrival process for a single server queue with memoryless services We assume E[S] = l/p = 1 and p = X For an Erlang-k interarrival time distribution with rate parameter kX, the mean interarrival time equals E[A] = Ic/lcX = l/X and the Laplace transform equals
(7.16)
Trang 77.2 The GIG11 queue 139
1.4 1.2 1.0 Fw(t) O-8
0.6 0.4 0.2 0.0 /
;;lv(t) -
t
Figure 7.1: The waiting time distribution in a Hypo-2IMIl queue (p = 0.667) The fixed-point iteration to solve becomes
(7.17)
In Figure 7.2 we show the value of 0 as a function of p for various values of k As can
be observed, we always have 0 5 p Moreover, the larger k, the more D deviates from p Hence, for a given utilisation p in the EklMll queue, a larger k implies a smaller utilisation
cr in the MIMI1 q ueue Clearly, increasing k removes variance from the model and thus
For the GI Cl 1 queue, explicit results for mean performance measures are even more difficult
to obtain The main difficulty lies in the complex state space of the stochastic process un- derlying this queueing system which, apart from the discrete number of customers present, also consists of continuous components for both the remaining service time and the remain- ing interarrival time Even an embedding approach as has been followed for the MlGll and the G (M I 1 queue is therefore not possible anymore Still though, an important gen- eral result known as Lindley’s integral equation can be obtained quite easily; its practical applicability is, however, limited
Trang 8140 7 GIMIl-FCFS and GIGIl-FCFS queueing models
1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
P
Figure 7.2: Values of o as a function of p for k = 1,2,9 (top to bottom) in the Ek[M[l queue
As before, we assume that we deal with independent and identically distributed inter- arrival and service times, with distributions (and densities) J”(t) (fA(t)) and R’s(t) (fs(t)), respectively Moments and (squared) coefficients of variation are denoted as usual Let us now try to express the waiting time perceived by the (n + 1)-th customer in terms of the waiting time of the n-th customer For that purpose, let r, denote the arrival time for the n-th customer, S, the service time and W, the waiting time perceived by the n-th customer Furthermore, we can define A, = r, - 7,-r, so that A, is the interarrival time between the n-th and the (n - l)-th arrival Notice that the random variables Sn and A, are in fact independent of n; they are only governed by the interarrival and service time distributions We have to distinguish two cases now (see also Figure 7.3):
(a) the (n + 1)-th customer finds a busy system: the sum of the service and waiting time of the n-th customer is more than the time between the arrival of the n-th and (n + 1)-th customer, and we have
W n+l = W, + Sn - A,+l, if Wn + Sn 2 &+I;
(b) the (n + l)-th customer finds an empty system: the sum of the service and waiting time of the n-th customer is less than the time between the arrival of the n-th and
Trang 97.2 The GIG11 queue 141
departures
jl-! J
server
It
I
arrivals
( ->I
A n+l W n+l I I
n n+l
(4
n-l
t( -
Gt
(b)
Figure 7.3: Two cases for the evolution of a GIG11 system: (a) system non-empty upon an arrival, and (b) system empty upon an arrival
(n + l)-th customer, and we have
W n+l = 0, if Wn + Sn 2 &+I-
The equations for these two cases describe the evolution of the GIG11 system By intro- ducing a new random variable U, = S, - An+l, we can rewrite them as
W n+l = maX{W, + Un, 0) =
{
Wn + un, Wn+Un L O,
o
’ Wn+Un 50 (7.18)
The random variable U, measures the difference in interarrival and service time of the n-th and (n + l)-th customer For stability of the GIG11 q ueue, it should have an expectation smaller than 0, meaning that, on average, the interarrival time is larger than the service time If we know the distribution of U,, we can calculate the distribution of W, To start,
we have to compute
Since u can be both negative and positive, we have to distinguish between these two cases
In Figure 7.4 we show the two possible cases On the x- and y-axis we have drawn S and
A (since our arguments are valid for all n, we can drop the subscript n) If u 2 0, the area that signifies the events “S-A < u” is the shaded area in Figure 7.4(a) over which we have
to integrate If u 5 0 we have to integrate over the shaded area in Figure 7.4(b) Since
Trang 10142 7 GlMl l-FCFS and GI GI l-FCFS queueing models
S
04
Figure 7.4: The two cases to be distinguished when computing Fu(u)
A and S are independent random variables, we have fA,s(a, s) = fA (a)fs (s) In summary,
we have
Pr{U 5 u} = so” so” fS(s).fA(+a ds + s,” Js:, f&)fA(a)da ds, U 2 0,
~oi h;, fds).fA(a)da ds, u 5 0
(7.20)
Note that in case u 2 0, the first integral reduces to J’s(u)
Now that we have obtained the distribution of U, we can derive the waiting time distribution Fw,(t) Clearly, we have FwJt) = 0, for t 5 0 For t > 0, we have from (7.18):
F&+1 (t) = Pr{W,+l 5 t} = Pr{W, + U, 5 t}
=
J 0 O” Pr{U, 5 t - w]IV, = w}dF~,,(w), (7.21) where we have conditioned on IV, taking a specific value w, which happens to be the case with probability density dF~,(w) We omit the subscripts n; this can be done under the assumption that the system is stable, i.e.,p < 1, in which case, in steady state, all customers experience the same waiting time distribution Because the random variables
U = S - A and W are independent, we can rewrite the above conditional probability as follows:
Fw(t) = /” Pr{U 5 t - w}dFw(w) = Jm FU(t - w)dFw(w)
Combining the above two results, we obtain Lindley’s integral equation:
(7.22)