ps5 - Problem Set ff5 Issued: November 8,2006 Due: November...

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Problem Set ff5 Issued: November 8,2006 Due: November 20,2006 Problem 1 Consider a I-mile-by-Smileh~opeou~ sertice regonsend by two mobilepatrollin3 sewers as shown in Figure I. Hm are the asswnptiom ofthe model: 1. Cwtoener locations are unifocmly bwtly located over the entire mtanplar ece region. 2. Chw time, customers arrive as a homo~meousPoisson process at aggrqate rate R=2 customers per how. 3, Uh k not busy sming a~stmers. each m-er patrols its sector (sector 1 or 2, rpqxctivel), Under tlus cirammce, the swer's location at a random tune is u~llformly dimh1ted OVH sector. Each sector is a one-mile-brae-mle sq~lare. 4, Tmd distance is n&t-arlgk ('?vlanhattanm&c--), 111th speed equal to 1000 rni.ihr. 5. The on-scale time to me a cr~tomer 1s a randomvanable ha\-ins a negative exponential probability density function with mean equal to 30 minutes. Upon completionof senice of a customer at the scene, the sm7erresumes random patrol of hdiw sector. 6. This. is the dispatch strate37 For a customer from i (*1,2), dispatch senrer i if anilable. Else &watch tile other sen= ~f ax~~lable; Else the is lost forever- x = -1 x= 0 x=I Sector 2 Sector 1 -x Patrolling senm %!
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(a) Is d true tlmt the ~~mkload (fraction of me hsv) of each sewer 1s eq11al to 1/2? If me, hefly explain why. If false, derive the cma figure. (b) Determine the fraction of dispatches that take sen~er 1 to sector 2. (c) Determine the mean travel time to a randorn sewed customer. Now wnsick the situation as shown in Figure Assumptions 1 through 5 abw main correct. Hor3.etTerer Assumption 6 is al- as follows: Cm in luhicl~ botp'l se~vws are azwiiable: For a custom fiom a part sector i not m Buffer Zone i, dspatch smw i For a customer Buffer i, &.;patch the other sen= only if that athw smw is with its mtm buffer zone and mer i within buffer zone; else dispatch smm i that customer. Cme in 1vhic11 on!v
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ps5 - Problem Set ff5 Issued: November 8,2006 Due: November...

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