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3 Arrival and departure functions at a typical queue. The area from 0 to time t between the curves a(t) and d(t) in Fig. 3 is the integral over time of the quantity N (t) and obviously represents the time spent by all the customers in the system (measured in units of customer-seconds) up to point t. Denote this area by I(t). 1 Little’s Law 63 1 .. 4 Simplified computer system used in the exercise Assuming that the limits λ = T = lim λt t→∞ lim Tt t→∞ exist, then so will the limit N for Nt the average number of customers in the system.

We already mentioned in Sec. 2 on page 26 that the time a Markov process spends in any state has to be memoryless. In the case of a DTMC this means that the chain must have geometrically distributed state sojourn times while a CTMC must have exponentially distributed sojourn times. This is such an important property that we include a proof from Kleinrock[108] of it here. The proof is also instructive in itself. Let yi be a random variable which describes the time spent in state i. The Markov property specifies that we may not remember how long we have been in state i which means that the remaining sojourn time in i may only depend upon i.

5 which illustrates the state transition diagramme of the birth-death process. Concentrating on state k we observe that one may enter it only from the state k − 1 or from the state k + 1 and similarly one leaves state k only to enter state k − 1 or state k + 1. From that diagramme it is clear why we refer to the process we described as the nearest-neighbour, birth-death process. The clever thing is to note that we can obtain Eqs. 6) directly from the state transition diagramme in Fig. 5 by equating the rates of flow into and out of each state k, k = 0,1, .

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