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Unformatted text preview: IEOR 6711: Stochastic Models I, Fall 2003, Professor Whitt Final Exam: Thursday, December 18, Chapters 4,5 and 9 in Ross Below are six questions with several parts. Do as much as you can. Show your work. 1. Two-Pump Gas Station (12 points) Potential customers arrive at a full-service, two-pump gas station according to a Poisson process at a rate of 40 cars per hour. There are two service attendants to help customers, one for each pump. If the two pumps are busy, then arriving customers wait in a single queue, to be served in the order of arrival by the first available pump. However, customers will not enter the station to wait if there are already two customers waiting, in addition to the two in service. Suppose that the amount of time required to service a car is exponentially distributed with a mean of three minutes. (a) In the long-run, what rate are cars served? (b) In the long-run, what proportion of potential customers fail to be served? (c) What is the long-run proportion of time that the station is empty? (d) In steady state, what is the probability that both pumps are busy? 2. System Failure in a Call Center (14 points) Consider a call center with 100 agents answering telephone calls, modelled as an Erlang delay system, i.e., as an M/M/ 100 / ∞ queue with Poisson arrival process having arrival rate λ , IID exponential service times each with mean 1 /μ , 100 homogenous servers (agents) and unlimited waiting space. Suppose that at some instant there is a system failure, which shuts down the call arrival process, but otherwise does not affect the system. Hence after that failure instant, there are no more arrivals, but the customers being served continue to receive their service. Suppose that there are 97 customers in service at the failure instant (and thus none waiting in queue). Let X ( t ) be the number of customers still in the system receiving service t time units after the failure event....
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This note was uploaded on 01/05/2010 for the course ECE 01 taught by Professor All during the Spring '09 term at Aarhus Universitet.
- Spring '09