Page 145 - A First Course In Stochastic Models

EXERCISES 137

to verify that
 
∞ ∞ ℓ
∞
t −µt (µt)
p j = π k  e  a(t) dt, j ≥ 1.
0 ℓ + 1 ℓ!
k=j−1 ℓ=k+1−j
(Hint: fix j and assume that the process incurs a cost at rate 1 whenever j customers are
present and a cost at rate 0 otherwise. Imagine that the server continues servicing fictitious
customers when the system is empty so that actual or fictitious service completions occur
according to a Poisson process with rate µ.)
3.19 In each time unit a job arrives at a conveyor with a single workstation. The workstation
can process only one job at a time and has a buffer with ample capacity to store the arriving
jobs that find the workstation busy. The processing times of the jobs are independent random
variables having a common Erlang (r, µ) distribution. It is assumed that r/µ < 1.
(a) Define an appropriate Markov chain to analyse the number of jobs in the buffer just
prior to the arrival epochs of new jobs and specify the one-step transition probabilities.
(b) Explain how to calculate the long-run average delay in the buffer per job.
(c) Prove that the equilibrium distribution of this Markov chain has a geometric tail.
3.20 Consider Exercise 3.19 again but now assume that the buffer has finite capacity. Any
arriving job that finds the buffer full is lost. Show how to calculate the long-run fraction
of lost jobs and the long-run fraction of time the workstation is busy (Hint: use Little’s
formula for the latter performance measure).
3.21 At the telephone exchange, calls arrive according to a Poisson process with rate λ.
The calls are first put in an infinite-capacity buffer before they can be processed further.
The buffer is periodically scanned every T time units, and only at those scanning epochs
are calls in the buffer allocated to free transmission lines. There are c transmission lines
and each transmission line can handle only one call at a time. The transmission times of
the calls are independent random variables having a common exponential distribution with
mean 1/µ.
(a) Use Markov chain analysis to find the equilibrium distribution {π j } of the number of
calls in the buffer just prior to the scanning epochs.
(b) Argue that the long-run average number of calls in the buffer is given by

∞
1
L q = (j − c)π j + λT.
2
j=c+1
(Hint: imagine that each call is marked upon arrival and is unmarked at the next scanning
epoch. Argue that the average number of marked calls in the buffer is 1 2 λT .)
(c) What is the long-run average delay in the buffer per call?
3.22 Consider Example 3.4.1 with Poisson arrivals of messages.

c−1
∞
π
(a) Prove the validity of the relation λ = jπ j + c j=c j and note that this
j=1
relation can be used as an accuracy check on the calculated values of the state probabilities
π j , j = 0, 1, . . . .
(b) Use the hint in Exercise 3.21 to prove that the long-run average number of messages
1
in the buffer equals
j=c+1 (j − c)π j + λT .
∞
2
(c) What is the long-run average delay in the buffer per message?
3.23 Consider Example 3.4.1 again but assume now that the buffer for temporarily storing
arriving messages has a finite capacity K. Each arriving message that finds the buffer full
is lost.