The Distributional Form of Littles Law And the Fuhrmann Cooper Decomposition
The Distributional Form of Littles Law And the Fuhrmann Cooper Decomposition
Julian Keilson
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1 -Peff«teef(s) and the transform of the waiting time is (2. 10) CL^is) - '^ 71b([X-S)A]). 1 -PEFFttTEEF^s) Proof: Equations (2. 8)-(2. 10) follow from (1. 1), (2. 2) and (2. 3).^ page 6 3. Special cases. CASE 1: M/G/1 Queue Here ot^-^ppCs) = (tj-Cs) and Nb = so that from (2. 3) (l-p)(l-u) (3. 1) Kq(u) = 'J"': / . CASE 2: M/G/1 with vacations and exhaustive service For this discipline, e. G. , [1], an M/G/1 queue is served exhaustively. Then the server is inactive, (i. E, "on vacation") for a d...uration V. At the end of a vacation period another vacation period begins if the system is empty. Otherwise the queue is again served exhaustively. It is assumed that V is independent of the arrival process. Again, Teff = T, the service time of the queue. If no service is in progress then the system must be on vacation. Nb is then equal to the number of customers that have arrived since the last vacation began. But the time at ergodicity since the last vacation began and the forward recurrence time V* of a vacation time are equal in distribution.
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