[4] Consider the BIZEEBEE airline that has a reservations clerk taking calls. If

Question

[4] Consider the BIZEEBEE airline that has a reservations clerk taking calls. If the clerk is busy the caller is put on hold. The calls are taken in the order received. Assume that calls arrive exponentially (Ca2 = 1) at the rate of one every 6 minutes. The clerk takes on average 5 minutes to complete the reservation. The time for service is also assumed to be exponentially distributed (Cs2 = 1). The clerk is paid $20 per hour. It has been estimated that each minute that a customer spends on hold costs BIZEEBEE $2 due to customer dissatisfaction and loss of future business. Estimate the average hourly cost.

(Hint: First calculate wait time, CTq, using available queueing formulae and then use the available information to calculate average hourly cost. Note that both interarrival times and service times are exponentially distributed.)

Explanation / Answer

The inter-arrival time of customers is 6 minutes and this follows an exponential distribution.

The service interval of the clerk is 5 minutes per customer.

This means that the queue follows M/M/1 model.

The arrival rate per hour is 60/6 = 10 customers (lambda)

The service rate per hour is 60/5 = 12 customers (mu)

Since the service rate is higher than the arrival rate, there will not be an infinite queue and the clerks will be free sometimes.

Under queueing theory, the waiting time in the queue is calculated using

Wq = lambda/[mu*(mu – lambda)]

Wq = 10/[12*(12-10)] = 10/[12*2] = 10/24 = 5/12 hours

This means that the average waiting time in queue is 5/12 hours or 25 minutes. This means that each customer waits on an average of 25 minutes in the queue.

The average length of the system is calculated by

Ls = lambda / (mu – lambda) = 10 / (12-10) = 5

Cost = cost of waiting per hour per customer * average customer in the queue + cost of the server per hour

Cost = 2*60*5 + 20 = 620

The average hourly cost is $620

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