THE POISSON PROCESS

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Then the estimate for k is 18450 Substituting this value into the square-root formula for ci , we nd c1 1583, c2 3423, c3 4592, c4 6305 and c5 9098 This suggests the allocation

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(c1 , c2 , c3 , c4 , c5 ) = (16, 34, 46, 63, 91)

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Note that in determining this allocation we have used the distributions of the processing times only through their rst moments The actual value of the long-run fraction of time during which all vehicles are occupied in region i depends (to a slight degree) on the probability distribution of the processing time Si Using simulation, we nd the values 0056, 0058, 0050, 0051 and 0050 for the service level in the respective regions 1, 2, 3, 4 and 5 The M/G/ queue also has applications in the analysis of inventory systems Example 114 A two-echelon inventory system with repairable items Consider a two-echelon inventory system consisting of a central depot and a number N of regional bases that operate independently of each other Failed items arrive at the base level and are either repaired at the base or at the central depot, depending on the complexity of the repair More speci cally, failed items arrive at the bases 1, , N according to independent Poisson processes with respective rates 1 , , N A failed item at base j can be repaired at the base with probability rj ; otherwise the item must be repaired at the depot The average repair time of an item is j at base j and 0 at the depot It takes an average time of j to ship an item from base j to the depot and back The base immediately replaces a failed item from base stock if available; otherwise the replacement of the failed item is back ordered until an item becomes available at the base If a failed item from base j arrives at the depot for repair, the depot immediately sends a replacement item to the base j from depot stock if available; otherwise the replacement is back ordered until a repaired item becomes available at the depot In the two-echelon system a total of J spare parts are available The goal is to spread these parts over the bases and the depot in order to minimize the total average number of back orders outstanding at the bases This repairable-item inventory model has applications in the military, among others An approximate analysis of this inventory system can be given by using the M/G/ queueing model Let (S0 , S1 , , SN ) be a given design for which S0 spare parts have been assigned to the depot and Sj spare parts to base j for j = 1, , N such that S0 + S1 + + SN = J At the depot, failed items arrive according to a Poisson process with rate

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Each failed item arriving at the depot immediately goes to repair The failed items arriving at the depot can be thought of as customers arriving at a queueing system

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THE POISSON PROCESS AND RELATED PROCESSES

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with in nitely many servers Hence the limiting distribution of the number of items in repair at the depot at an arbitrary point of time is a Poisson distribution with mean 0 0 The available stock at the depot is positive only if less than S0 items are in repair at the depot Why Hence a delay occurs for the replacement of a failed item arriving at the depot only if S0 or more items are in repair upon arrival of the item De ne now W0 = the long-run average amount of time a failed item at the depot waits before a replacement is shipped, L0 = the long-run average number of failed items at the depot waiting for the shipment of a replacement A simple relation exists between L0 and W0 On average 0 failed items arrive at the depot per time unit and on average a failed item at the depot waits W0 time units before a replacement is shipped Thus the average number of failed items at the depot waiting for the shipment of a replacement equals 0 W0 This heuristic argument shows that L 0 = 0 W0 This relation is a special case of Little s formula to be discussed in Section 23 The relation W0 = L0 / 0 leads to an explicit formula for W0 , since L0 is given by

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