From this it follows that xx y y y x so satises

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Unformatted text preview: the state times the amount of time we spend there. Similarly the last three are 35 ·1 60 16 1 · 60 2 91 · 60 3 where again these are the probability we visit the state times the amount of time we spend there. 137 4.5. MARKOVIAN QUEUES 4.5 Markovian Queues In this section we will take a systematic look at the basic models of queueing theory that have Poisson arrivals and exponential service times. The arguments in Section 3.2 explain why we can be happy assuming that the arrival process is Poisson. The assumption of exponential services times is hard to justify, but here, it is a necessary evil. The lack of memory property of the exponential is needed for the queue length to be a continuous time Markov chain. We begin with the simplest examples. Single server queues Example 4.23. M/M/1 queue. In this system customers arrive to a single server facility at the times of a Poisson process with rate , and each requires an independent amount of service that has an exponential distribution with rate µ. From the description it should be clear that the transition rates are q (n, n + 1) = q (n, n 1) = µ if n if n 0 1 so we have a birth and death chain with birth rates n = and death rates µn = µ. Plugging into our formula for the stationary distribution, (4.18), we have ✓ ◆n n 1 ··· 0 ⇡ (n) = · ⇡ (0) = ⇡ (0) (4.22) µn ·...
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