Page 232 - A First Course In Stochastic Models

EXERCISES 225

(a) Give a recursive relation for the computation of the equilibrium distribution {p j } of
the number of customers present.
(b) What is the long-run fraction of customers who are delayed? Can you explain why
∞

θ j=c (j − c)p j /λ gives the long-run fraction of customers who prematurely leave the
system?
5.4 An information centre provides service in a bilingual environment. Requests for ser-
vice arrive by telephone. Service requests of major-language customers and minor-language
customers arrive according to independent Poisson processes with respective rates λ 1 and
λ 2 . There are c bilingual agents to handle the service requests. Each service request find-
ing all c agents occupied upon arrival waits in queue until a free agent becomes available.
The service time of a major-language request is exponentially distributed with mean 1/µ 1
and that of a minor-language request has an exponential distribution with mean 1/µ 2 . Let
p(i, i 1 , i 2 ) denote the joint equilibrium probability that simultaneously i 1 agents are ser-
vicing major-language customers, i 2 agents are servicing minor-language customers and i
service requests are waiting in queue. Use the equilibrium equations of an appropriately
chosen continuous-time Markov chain and use generating functions to prove that for any
i 1 = 0, 1, . . . , c there is a constant γ (i 1 ) such that

p(i, i 1 , c − i 1 ) ∼ γ (i 1 )τ −i as i → ∞,

with τ = 1 + δ/λ, where λ = λ 1 + λ 2 and δ is the unique solution of
2 2
δ − (cµ 1 + cµ 2 − λ)δ + c µ 1 µ 2 − cλ 1 µ 2 − cλ 2 µ 1 = 0
on the interval (0, c min(µ 1 , µ 2 )).
5.5 Consider the following modification of Example 2.5.1. Overflow is allowed from one
loo to another when there is a queue at one of the loos and there is nobody at the other loo.
It is assumed that the occupation times at the loos are exponentially distributed. Formulate
a continuous-time Markov chain to analyse the new situation. Assume the numerical data
λ w = λ m = 0.6, µ w = 1.5 and µ m = 0.75. Solve the equilibrium equations and compare
the average queue sizes for the women’s loo and the men’s loo with the average queue sizes
in the situation of strictly separated loos.
5.6 Jobs of types 1 and 2 arrive according to independent Poisson processes with respective
rates λ 1 and λ 2 . Each job type has its own queue. Both queues are simultaneously served,
where service is only provided to the job at the head of the queue. If both queues are not
empty, service is provided at unity rate at each queue. A non-empty queue for type i jobs
receives service at a rate of r i ≥ 1 when the other queue is empty (i = 1, 2). The service
requirement of a type i job has an exponential distribution with mean 1/µ i . The service
requirements of the jobs are independent of each other. It is assumed that ρ i = λ i /µ i is
less than 1 for i = 1, 2. Let p(i 1 , i 2 ) be the joint equilibrium probability of having i 1 jobs
at queue 1 and i 2 jobs at queue 2. Set up the equilibrium equations for the probabilities
i 1 i 2
p(i 1 , i 2 ). Do numerical investigations to find out whether or not p(i 1 , i 2 ) ∼ γρ ρ 2 as
1
i 1 → ∞ and i 2 → ∞ for some constant γ .
5.7 Consider a production hall with two machines. Jobs arrive according to a Poisson process
with rate λ. Upon arrival a job has to be assigned to one of the two machines. Each machine
has ample waiting space for jobs that have to wait. Each machine can handle only one job
at a time. If a job is assigned to machine i, its processing time is exponentially distributed
with mean 1/µ i for i = 1, 2. The control rule is to assign an arriving job to the machine
with the shortest queue (if both queues are equal, machine group 1 is chosen). Jockeying of
the jobs is not possible. Use Markov-chain analysis to find the equilibrium probability that
the delay of a job in queue is longer than a given time t 0 .