Page 235 - A First Course In Stochastic Models

228 MARKOV CHAINS AND QUEUES

and solve for the numerical data λ = 1, 1/µ 1 = 2, 1/µ 2 = 5, s 1 = 5 and s 2 = 10. (Answer:
0.0464.) Verify experimentally that the loss probability is nearly insensitive to the distribu-
tional form of the service times (e.g. compute the loss probability 0.0470 for the above data
when the service time in group 1 has an E 2 distribution and the service time in group 2 has
an H 2 distribution with balanced means and a squared coefficient of variation of 4).
5.19 Customers of the types 1, . . . , m arrive at a service centre according to indepen-
dent Poisson processes with respective rates λ 1 , . . . , λ m . The service centre has c identical
servers. An arriving customer of type j requires b j servers and is lost when there are no b j
servers available. A customer of type j has an exponentially distributed service time with
mean 1/µ j for j = 1, . . . , m. The customer keeps all of the assigned b j servers busy during
his service time and upon completion of the service time the b j servers are simultaneously
released. Let p(n 1 , . . . , n m ) be the long-run fraction of time that n j groups of b j servers
are handling type j customers for j = 1, . . . , m.
(a) Verify from the equilibrium equations for the probabilities p(n 1 , . . . , n m ) that, for
some constant C > 0,
m
(λ j /µ j ) n j
p(n 1 , . . . , n m ) = C
n j !
j=1
for all (n 1 , . . . , n m ) with n 1 b 1 + · · · + n m b m ≤ c.
(b) What is the long-run fraction of type j customers who are lost?
The above product-form solution can also be proved by considering the process
{(X 1 (t), . . . , X m (t))} in the infinite-server model (c = ∞) with X j (t) denoting the number
of type j customers present at time t. The processes {X 1 (t)}, . . . , {X m (t)} are indepen-
dent of each other and each separate process {X j (t)} constitutes an M/M/∞ queueing
process having a Poisson distribution with mean λ j /µ j as equilibrium distribution. Noting
that the process {(X 1 (t), . . . , X m (t))} is time reversible, it can be concluded from the result
in Exercise 5.8 that the above product-form solution holds. The normalization constant C
can be computed as follows. Let {p j , 0 ≤ j ≤ c} denote the equilibrium distribution of
(∞)
the numbers of busy servers in the loss model with c servers and let {p } denote the
j
equilibrium distribution of the number of busy servers in the infinite-server model. Then
(∞)
p
j
p j = , j = 0, 1, . . . , c.
c (∞)
p
k=0 k
(∞)
The normalization constant C is given by p 0 . It is left to the reader to verify that {p } can
j
be computed as the convolution of m compound Poisson distributions. The jth compound
Poisson distribution represents the limiting distribution of the numbers of busy servers in a
X
batch arrival M /G/∞ queue with group service, where the arrival rate of batches is λ j ,
each batch consists of b j customers and the mean service time of the customers from the
same batch is 1/µ j ; see part (b) of Exercise 1.15. Finally, it is noted that the loss model
has the insensitivity property.
5.20 Batches of containers arrive at a stockyard according to a Poisson process with a rate
of λ = 15 batches per day. Each batch consists of two or three containers with respective
1
probabilities of 2 3 and . The stockyard has space for only 50 containers. An arriving batch
3
finding not enough space is lost and is brought elsewhere. Containers from the same batch
are removed simultaneously after a random time. The holding times of the batches are
independent random variables and have a lognormal distribution with a mean of 1 day and
a standard deviation of 2 days for batches of size 3 and a mean of 1 day and a standard
deviation of 1 2 day for batches of size 3. Calculate the long-run fraction of batches of size
2 that are lost and the long-run fraction of batches of size 3 that are lost.