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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
µn xn 1
f (x) =
Changing variables m = n 1 and rearranging, the above becomes
X m xm
)e (µ )x
m=0 Recalling that P (TQ > 0) = /µ, we can see that the last result says that
the conditional distribution of TQ given that TQ > 0 is exponential with rate
. 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:
WQ = E (s2 /2)
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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