04quiz2 - Massachusetts Institute of Technology Logistical...

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Massachusetts Institute of Technology Logistical and Transportation Planning Methods 1.203J/6.281J/15.073J/16.76J/ESD.216J Quiz #2 OPEN BOOK December 6, 2004 Please do Problems 1 and 2 in one booklet and Problem 3 in a separate one. Remember to put your name on each booklet! And please explain all of your work! Good luck! 1
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Problem 1 (30 points): Queuing in Pairs [Note: This problem is a variation of Problem # 4 in Quiz 1. Several things have, however, changed in this problem: Type 2 customers are described differently; the queueing capacity of the system is 2 instead of 0; and some of the questions are different, as well.] Consider a queueing system with two parallel servers and two spaces for queueing (in addition to the two servers). This facility serves two types of customers. Type 1 customers are of the conventional type. They arrive in a Poisson manner at a rate of λ 1 per minute. The service time to these customers has a negative exponential pdf with a rate of µ 1 per minute for each server. Any arriving Type 1 customers who find the system full (i.e., 4 customers in the system) are lost. Type 2 customers are unusual. They, too, arrive in a Poisson manner at a rate of 2 per minute, but they arrive IN PAIRS. [Think of a restaurant, where some of the customers (Type 1) arrive individually, and others (Type 2) arrive in pairs.] Moreover, when service to each one of these pairs begins, the pair occupies simultaneously TWO servers (i.e., both of the servers). The servers work together on each of these Type 2 pairs. The service time to the pair has a negative exponential pdf with a rate of 2 per minute. (Note that this means that the two servers begin and end service to each Type 2 pair simultaneously and, together, can serve 2 Type 2 pairs per minute if working continuously on Type 2 pairs.) Type 2 customers who do not find at least TWO available spaces upon arrival are lost. [Please note: A Type 2 pair cannot occupy the servers, unless both servers are available. Thus, in the case where one Type 1 customer is in service and the only customers waiting are one Type 2 pair, the Type 2 pair must still wait in queue and the second server remains idle.] (a) (20 points) Please draw carefully a state transition diagram that describes this queueing system. Please make sure to define clearly the states of the system. [Note: You can answer part (b) without answering part (a); but, it will be easier to answer (b), if you have answered (a).] (b) (10 points) Suppose that there are currently four Type 1 customers in the system. (Obviously, two of them are receiving service and the other two are in queue.)
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04quiz2 - Massachusetts Institute of Technology Logistical...

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