Page 430 - A First Course In Stochastic Models

EXERCISES 425

9.12 Use exact results from Section 9.5.3 to verify numerically that

∞
app 1 − B(D) app D (t − D)b(t) dt
P and W q =
delay =
∞ δ(t−D)
∞ δ(t−D)
D e b(t) dt B(D) − 1 + D e b(t) dt
are excellent approximations to P delay and W q in the D/G/1 queue. Here B(t) and b(t)
are the probability distribution function and the probability density of the service time. The
constant δ is the unique positive solution to e −δD
∞ δy
e b(y) dy = 1. These approximations
0
for the D/G/1 queue are due to Fredericks (1982).
9.13 Consider the machine-repair model from Exercise 5.2. Assume now that the service time
S of a request has a general probability distribution function B(x). Extend the approximate
analysis of the M/G/c queue in Section 9.6.2 to the machine-repair model. Verify that the
resulting approximation to the limiting distribution {p j } of the number of service requests
in the system is given by
app N j app
p = ( )[νE(S)] p , 0 ≤ j ≤ c − 1
j j 0
j
app app app
p = (N − c + 1)να cj p + (N − k)νβ kj p , c ≤ j ≤ N
j c−1 k
k=c
with
∞ ∞

c−1
α cj = {1 − B e (t)} {1 − B(t)}φ cj (t) dt, β kj = {1 − B(ct)}φ kj (t) dt,
0 0
where B e (t) denotes the equilibrium excess distribution of B(t) and φ kj (t) is given by
N−k −νt j−k −νt(N−j)
" #
φ kj (t) = (1 − e ) e , t > 0 and k ≥ j ≥ c.
j−k
9.14 Consider the finite-capacity M/D/c/c + N queue with deterministic services. It is
assumed that the server utilization is less than 1. Let W q (x) be the limiting distribution of
the delay in queue of an accepted customer. For k = 1, . . . , c, let
c !
j
c " x # " x # c−j
U k (x) = 1 − , 0 ≤ x ≤ D
j D D
j=k
be the probability distribution function of the kth order statistic of c independent random vari-
ables that are uniformly distributed on (0, D). An approximation to W q (x) can be calculated
by the following algorithm:
app
Step 0. Use the results of Theorem 9.8.1 to compute approximations p to the state
j
probabilities p j in the M/D/c/c + N queue.
N+c−1 app app

Step 1. Approximate 1 − W q (x) by [p /(1 − p )]V j (x), where V kc+r (x) is
j=c j N+c
given by 1 − U r+1 (x − kD) for k ≥ 0 and 0 ≤ r ≤ c − 1.
Use computer simulation to find out how well this approximation to W q (x) performs.
Investigate the quality of the approximation to W q (x) which results by approximating p j
(∞)
through γp ∞ for 0 ≤ j ≤ N + c − 1 in accordance with (9.8.3), where p is the state
j j
probability in the M/D/c queue. Further, investigate how well the two-moment approxima-
tion (9.6.31) works for the conditional waiting-time percentiles in the M/G/c/c + N queue
(the computation of W q (x) in the M/M/c/c + N queue is discussed in Exercise 5.1).
9.15 Consider a single-server queueing system in which the arrival process is the result of
the superposition of m homogeneous on-off sources. Each source is alternately on and off,
where the on-time has an exponential distribution with mean 1/ν on and the off-time has
an exponential distribution with mean 1/ν off . The sources act independently of each other.