Page 188 - A First Course In Stochastic Models

EXERCISES 181

(b) What is the long-run fraction of time the channel is idle? What is the long-run fraction
of messages that are rejected?
(c) What is the long-run average number of messages waiting to be transmitted? What is
the long-run average delay in queue per accepted message?
4.9 An information centre has one attendant; people with questions arrive according to a
Poisson process with rate λ. A person who finds n other customers present upon arrival
joins the queue with probability 1/(n + 1) for n = 0, 1, . . . and goes elsewhere otherwise.
The service times of the persons are independent random variables having an exponential
distribution with mean 1/µ.
(a) Verify that the equilibrium distribution of the number of persons present has a Poisson
distribution with mean λ/µ.
(b) What is the long-run fraction of persons with requests who actually join the queue?
What is the long-run average number of persons served per time unit?
4.10 (a) Consider Exercise 4.1 again. Specify the equilibrium equations for the state prob-
abilities. What is the long-run average waiting time of a carried passenger? What is the
long-run fraction of potential customers who are lost?
(b) Answer the questions in (a) again for the modified situation in which a potential
customer only waits when, upon his arrival, a sheroot is present.
4.11 Consider Exercise 4.2 again and denote by S ij the time needed to serve an alarm for
district j by unit i. Assume that S ij has a Coxian-2 distribution for all i, j. Show how to
calculate the following performance measures: π L = the fraction of alarms that is lost and
P i = the fraction of time that unit i is busy for i = 1, 2. Letting m ij and c 2 denote the
ij
mean and the squared coefficient of variation of S ij , assume the numerical data λ 1 = 0.25,
λ 2 = 0.25, m 11 = 0.75, m 12 = 1.25, m 21 = 1.25 and m 22 = 1. Write a computer program
to verify the following numerical results:
2 1
(i) π L = 0.0704, P 1 = 0.2006, P 2 = 0.2326 when c = for all i, j;
ij 2
2
(ii) π L = 0.0708, P 1 = 0.2004, P 2 = 0.2324 when c = 1 for all i, j;
ij
2
(iii) π L = 0.0718, P 1 = 0.2001, P 2 = 0.2321 when c = 4 for all i, j.
ij
2 1
Here the values c = , 1 and 4 correspond to the E 2 distribution, the exponential distri-
ij 2
bution and the H 2 distribution with balanced means.
4.12 In an inventory system for a single product the depletion of stock is due to demand
and deterioration. The demand process for the product is a Poisson process with rate λ. The
lifetime of each unit product is exponentially distributed with mean 1/µ. The stock control
is exercised as follows. Each time the stock drops to zero an order for Q units is placed. The
lead time of each order is negligible. Determine the average stock and the average number
of orders placed per time unit.
4.13 Messages arrive at a communication channel according to a Poisson process with rate
λ. The message length is exponentially distributed with mean 1/µ. An arriving message
finding the line idle is provided with service immediately; otherwise the message waits until
access to the line can be given. The communication line is only able to submit one message
at a time, but has available two possible transmission rates σ 1 and σ 2 with 0 < σ 1 < σ 2 .
Thus the transmission time of a message is exponentially distributed with mean 1/(σ i µ)
when the transmission rate σ i is used. It is assumed that λ/(σ 2 µ) < 1. At any time the
transmission line may switch from one rate to the other. The transmission rate is controlled
by a rule that uses a single critical number. The transmission rate σ 1 is used whenever less
than R messages are present, otherwise the faster transmission rate σ 2 is used. The following
costs are involved. There is a holding cost at rate hj whenever there are j messages in the
system. An operating cost at rate r i > 0 is incurred when the line is transmitting a message
using rate σ i , while an operating cost at rate r 0 ≥ 0 is incurred when the line is idle.