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(a) Use the uniformization method to compute E( ), ( ) and P { > t} for t = 2, 5 and 10 when = 1, = 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 418 Messages arrive at a node in a communication network according to a Poisson process with rate Each arriving message is temporarily stored in an in nite-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/ i (i = 1, 2), where 1/ 1 < 1/ 2 and 1 + 2 > 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) 419 Two communication lines in a packet switching network share a nite 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 1 and 2 A message of type j is destined for communication line j and its transmission time is exponentially distributed with mean 1/ 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 Nj of buffer places are reserved for messages of type j and a number N0 of buffer places are to be used by messages of both types, where N0 + N1 + N2 = M That is, an arriving message of type j is accepted only when the buffer is not full and less than N0 + N1 other messages of the same type j are present; otherwise, the message is rejected Discuss how to calculate the optimal values of N0 , N1 and N2 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, 1 = 2 = 1 and 1 = 2 = 1 This problem is based on Kamoun and Kleinrock (1980) 420 A traf c source is alternately on and off, where the on- and off-times are exponentially distributed with respective means 1/ and 1/ During on-periods the traf c source generates messages for a transmission channel according to a Poisson process with rate 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/ The on-times, off-times and transmission times are independent of each other Further, it is assumed that /[ ( + )] < 1 Let the states (i, 0) and (i, 1) correspond to the situation that there are i messages at the transmission channel and the traf c source is off or on respectively (a) Verify for the numerical values = 1, = 1, = 2, = 05 that the system of linear equations (446) is given by 1 3z 2z 05z z2 25z + 1 G0 (z) G1 (z) = (1 z)p00 (1 z)p01.

Verify the roots of det A(z) = 0 are z0 = 1, z1 = 02712865 and z2 = 12287136 (b) Use the roots z0 and z1 and the fact that Gi (z) is analytic for |z| 1 to nd p00 and p01

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