7 1 e si the so called pollaczek khintchine formula

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: e that customers pay $1 per minute in the queue and repeat the derivation of Little’s formula, then LQ = (3.4) a WQ The length of the queue is 1 less than the number in the system, except when the number in the system is 0, so if ⇡ (0) is the probability of no customers, then LQ = L 1 + ⇡ (0) Combining the last three equations with our first cost equation: ⇡ (0) = LQ (L 1) = 1 + a (WQ W) = 1 a Esi (3.5) Recalling that Esi = 1/µ, we have a simple proof that the inequality in Theorem 3.5 is sharp. 108 3.2.3 CHAPTER 3. RENEWAL PROCESSES M/G/1 queue Here the M stands for Markovian input and indicates we are considering the special case of the GI/G/1 queue in which the inputs are a rate Poisson process. The rest of the set-up is as before: there is a one server and the ith customer requires an amount of service si , where the si are independent and have a distribution G with mean 1/µ. When the input process is Poisson, the system has special properties that allow us to go further. We learned in Theorem 3.5 that if < µ then a GI/G/1 queue will repeatedly return to...
View Full Document

Ask a homework question - tutors are online