# We imagine that customers arrive at the times of a

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Unformatted text preview: re are n people in the system when 138 CHAPTER 4. CONTINUOUS TIME MARKOV CHAINS the customer arrives, then the amount of time he needs to enter service has a gamma(n, µ) density, so using (2.12) in Chapter 2 ◆ ✓ ◆n 1 X✓ µn xn 1 f (x) = 1 e µx µ µ (n 1)! n=1 Changing variables m = n 1 and rearranging, the above becomes ✓ ◆ 1 X m xm =1 e µx = (µ )e (µ )x µ m! µ m=0 Recalling that P (TQ &gt; 0) = /µ, we can see that the last result says that the conditional distribution of TQ given that TQ &gt; 0 is exponential with rate µ . From this we see that WQ = ETQ = 1 · µµ To compare with the Pollaczek-Khintchine formula, (3.7), we note that the service times si have Es2 /2 = 1/µ2 to conclude: i WQ = E (s2 /2) /µ2 1 i = =· 1 E si 1 /µ µµ With the waiting time in the queue calculated, we can see that the average waiting time in the system is W = WQ + Esi = 1 · µµ + 1µ · µµ = 1 µ To get this result using Little’s formula L = W we note that the queue length...
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## This document was uploaded on 03/06/2014 for the course MATH 4740 at Cornell University (Engineering School).

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