12.08.Spring

# 12.08.Spring - ISyE 3232A/C Spring 2008 Homework 12 Will...

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ISyE 3232A/C - Spring 2008 Homework 12 Will not be collected Homework 12 1. Customers arrive at a two-server system according to a Poisson process with rate λ = 10 per hour. An arrival finding server 1 free will begin his service with him. An arrival finding server 1 busy, server 2 free will join server 2. An arrival finding both servers busy goes away. Once a customer is served by either server, he departs the system. The service times at both servers are exponential random variables. Assume that the service rate of the first server is 6 per hour and the service rate of the second server is 4 per hour. (a) Describe a continuous time Markov chain to model the system and give the rate transition diagram. (b) Find the stationary distribution of the continuous time Markov chain (the long-run frac- tion of time for each state). (c) What is the long-run fraction of time that server i is busy, i = 1 , 2? 2. Consider a call center that is staffed by K agents with three phone lines. Call arrivals follow a Poisson process with rate 1 per minute. An arrival call that finds all lines busy is lost. That is, if K = 1,an arrival finding the system empty gets one phone line and service immediately. An arrival finding one customer in the system (being in service) gets the 2nd phone line waiting for the agent. An arrival finding two customers in the system (one in service and one waiting for the agent) gets the 3rd phone line. An arrival finding three customers in the system (one in service and two waiting for the agent) leaves the system because there is no available phone line. Call processing times are exponentially distributed with mean 2 minutes. (a) Find the throughput (the actual rate of customers being served or the actual rate of customers entered the system) and average waiting time when K = 1. (b) Find the throughput and average waiting time when K = 2. (c) Find the throughput and average waiting time when K = 3. 3. Consider a service system with a single server whose service time is exponentially distributed with rate μ and infinite capacity. The arrivals come to the system following a Poisson process of rate λ but an arrival finding i

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