3. () Telephone Exchange Queue. A telephone exchange consists of N operators on

Question

3. () Telephone Exchange Queue. A telephone exchange consists of N operators on N lines. Assume that the calling and servicing patterns are Poisson with p ak = a, .-k respectively for queue size of 0 k N and for sorne , equilibrium distribution for the queue is: ers he k! 1 N! Of particular interest, in this model, is the frequency with which the exchange becomes saturated over a long period of time i.e·Tx. This should be a small number for an efficient exchange since it provides a measurement of the number of calls lost to the system. (a) A telephone exchange has 5 operators who can handle 40 calls an hour each on average. If calls arrive at the exchange on average every 40 secs, show that there is approximately a 5.5% chance that the exchange will become saturated after a long time.

Explanation / Answer

3. () Telephone Exchange Queue. A telephone exchange consists of N operators on N lines. Assume that the calling and servicing patterns are Poisson with p ak = a, .-k respectively for queue size of 0 k N and for sorne , equilibrium distribution for the queue is: ers he k! 1 N! Of particular interest, in this model, is the frequency with which the exchange becomes saturated over a long period of time i.e·Tx. This should be a small number for an efficient exchange since it provides a measurement of the number of calls lost to the system. (a) A telephone exchange has 5 operators who can handle 40 calls an hour each on average. If calls arrive at the exchange on average every 40 secs, show that there is approximately a 5.5% chance that the exchange will become saturated after a long time.

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