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City Cab, Inc., uses two dispatchers to handle requests for service and to dispatch the cabs. The...

City Cab, Inc., uses two dispatchers to handle requests for service and to dispatch the cabs. The telephone calls that are made to City Cab use a common telephone number. When both dispatchers are busy, the caller hears a busy signal; no waiting is allowed. Callers who receive a busy signal can call back later or call another cab company for service. Assume that the arrival of calls follows a Poisson distribution, with a mean of 40 calls per hour, and that the call handling time follows an exponential probability distribution with a mean service time of 2 minutes. Based on this information, answer the following questions.

  1. What percentage of the time are both dispatchers idle?
  1. What percentage of the time are both dispatchers busy?
  1. What is the probability that a caller will receive a busy signal if 2, 3, or 4 dispatchers are used?
  1. If management wants no more than 12% of the callers to receive a busy signal, how many dispatchers should be used?

  1. Suppose the service time distribution is not exponential, it follows some other distribution such as a normal, but the mean service time remains at 2 minutes. Does this make any difference in the model used and in the results? Explain.
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Answer #1

Arrival rate = 40 calls per hour

Utilization = Arrival rate/service rate

= 40/60 = 0.6667

What percentage of the time are both dispatchers idle?

percentage of the time both dispatchers are idle = 1-Utilization = 1-0.6667 = 0.3333 = 33.33%

What percentage of the time are both dispatchers busy?

percentage of the time both dispatchers are busy = Utilization = 66.67%

What is the probability that a caller will receive a busy signal if 2, 3, or 4 dispatchers are used?

probability that a caller will receive a busy signal if 2 dispatchers are used = 40/60 = 0.6667

probability that a caller will receive a busy signal if 3 dispatchers are used = 40/(3*30) = 40/90 = 0.4444

probability that a caller will receive a busy signal if 4 dispatchers are used = 40/(4*30) = 40/120 = 0.3333

If management wants no more than 12% of the callers to receive a busy signal, how many dispatchers should be used?

If 11 dispatchers are used we can see utilization = 40/(11*30) = 12.12%

If 12 dispatchers are used we can see utilization = 40/(12*30) = 1/9 = 11.11%

So, 12 dispatchers should be used if management wants no more than 12% of the callers to receive a busy signal

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