Determining the Number of ServersIn the service department of the Glenn-Mark Auto Agency, mechanics requiring parts for auto repair or service present their request forms at the parts department counter. The parts clerk fills a request while the mechanic waits. Mechanics arrive in a random (Poisson) fashion at the rate of 40 per hour, and a clerk can fill requests at the rate of 20 per hour (exponential).If the cost for a parts clerk is $30 per hour and the cost for a mechanic is $60 per hour, determine the optimum number of clerks to staff the counter. (Because of the high arrival rate, an infinite source may be assumed.)

Determining the Number of ServersIn the service department of the Glenn-Mark Auto Agency, mechanics requiring parts for auto repair or service present their request forms at the parts department counter. The parts clerk fills a request while the mechanic waits. Mechanics arrive in a random (Poisson) fashion at the rate of 40 per hour, and a clerk can fill requests at the rate of 20 per hour (exponential).If the cost for a parts clerk is $30 per hour and the cost for a mechanic is $60 per hour, determine the optimum number of clerks to staff the counter. (Because of the high arrival rate, an infinite source may be assumed.)

Practical Management Science

6th Edition

ISBN:9781337406659

Author:WINSTON, Wayne L.

Publisher:WINSTON, Wayne L.

Chapter12: Queueing Models

Section12.5: Analytic Steady-state Queueing Models

Problem 21P

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The parts department of a large automobile dealership has a counter used exclusively for mechanics’ requests for parts. The time between requests can be modeled by a negative exponential distribution that has a mean of five minutes. A clerk can handle requests at a rate of 15 per hour, andthis can be modeled by a Poisson distribution that has a mean of 15. Suppose there are two clerksat the counter.a. On average, how many mechanics would be at the counter, including those being served?b. What is the probability that a mechanic would have to wait for service?c. If a mechanic has to wait, how long would the average wait be?d. What percentage of time are the clerks idle?e. If clerks represent a cost of $20 per hour and mechanics a cost of $30 per hour, what numberof clerks would be optimal in terms of minimizing total cost?
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