Page 190 - A First Course In Stochastic Models

EXERCISES 183

(a) Use the uniformization method to compute E(τ), σ(τ) and P{τ > t} for t = 2, 5 and
10 when λ = 1, µ = 10 and the number of standby units is varied as R = 1, 2 and 3.
(b) Extend the analysis in (a) for the case that the repair time has a Coxian-2 distribution
and investigate how sensitive the results in (a) are to the second moment of the repair-time
distribution.
4.18 Messages arrive at a node in a communication network according to a Poisson process
with rate λ. Each arriving message is temporarily stored in an infinite-capacity buffer until
it can be transmitted. The messages have to be routed over one of two communication lines
each with a different transmission time. The transmission time over the communication
line is i exponentially distributed with mean 1/µ i (i = 1, 2), where 1/µ 1 < 1/µ 2 and
µ 1 + µ 2 > λ. The faster communication line is always available for service, but the slower
line will be used only when the number of messages in the buffer exceeds some critical
level. Each line is only able to handle one message at a time and provides non-pre-emptive
service. With the goal of minimizing the average sojourn time (including transmission time)
of a message in the system, the following control rule with switching level L is used. The
slower line is turned on for transmitting a message when the number of messages in the
system exceeds the level L and is turned off again when it completes a transmission and
the number of messages left behind is at or below L. Show how to calculate the average
sojourn time of a message in the system. This problem is taken from Lin and Kumar (1984).
4.19 Two communication lines in a packet switching network share a finite storage space
for incoming messages. Messages of the types 1 and 2 arrive at the storage area according
to two independent Poisson processes with respective rates λ 1 and λ 2 . A message of type j
is destined for communication line j and its transmission time is exponentially distributed
with mean 1/µ j , j = 1, 2. A communication line is only able to transmit one message at
a time. The storage space consists of M buffer places. Each message requires exactly one
buffer place and occupies the buffer place until its transmission time has been completed.
A number N j of buffer places are reserved for messages of type j and a number N 0 of
buffer places are to be used by messages of both types, where N 0 + N 1 + N 2 = M. That
is, an arriving message of type j is accepted only when the buffer is not full and less than
N 0 + N 1 other messages of the same type j are present; otherwise, the message is rejected.
Discuss how to calculate the optimal values of N 0 , N 1 and N 2 when the goal is to minimize
the total rejection rate of both types of message. Write a computer program and solve for
the numerical data M = 15, λ 1 = λ 2 = 1 and µ 1 = µ 2 = 1. This problem is based on
Kamoun and Kleinrock (1980).
4.20 A traffic source is alternately on and off, where the on- and off-times are exponentially
distributed with respective means 1/δ and 1/β. During on-periods the traffic source gener-
ates messages for a transmission channel according to a Poisson process with rate λ. The
transmission channel can handle only one message at a time and the transmission time of a
message has an exponential distribution with mean 1/µ. The on-times, off-times and trans-
mission times are independent of each other. Further, it is assumed that λβ/[µ(δ + β)] < 1.
Let the states (i, 0) and (i, 1) correspond to the situation that there are i messages at the
transmission channel and the traffic source is off or on respectively.
(a) Verify for the numerical values λ = 1, µ = 1, β = 2, δ = 0.5 that the system of
linear equations (4.4.6) is given by

1 − 3z 0.5z G 0 (z) (1 − z)p 00 .
2
2z z − 2.5z + 1 G 1 (z) = (1 − z)p 01
Verify the roots of det A(z) = 0 are z 0 = 1, z 1 = 0.2712865 and z 2 = 1.2287136.
(b) Use the roots z 0 and z 1 and the fact that G i (z) is analytic for |z| ≤ 1 to find p 00 and
p 01 .