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Unformatted text preview: n the M/G/1 queue. We deﬁne the workload
in the system at time t, Zt , to be the sum of the remaining service times of all
customers in the system, and deﬁne the long run average workload to be
1
Z = lim
t!1 t Z t Zs ds 0 As in the proof of Little’s formula we will derive our result by computing
the rate at which revenue is earned in two ways. This time we suppose that
each customer in the queue or in service pays at a rate of $y when his remaining
service time is y ; i.e., we do not count the remaining waiting time in the queue.
If we let Y be the average total payment made by an arriving customer, then
our cost equation reasoning implies that the average workload Z satisﬁes
Z= Y 109 3.3. AGE AND RESIDUAL LIFE* Since a customer with service time si pays si during the qi units of time spent
waiting in the queue and at rate si x after x units of time in service
✓Z si
◆
Y = E (si qi ) + E
si x dx
0 Now a customer’s waiting time in the queue can be determined by looking
at the arrival process and at the service times of previ...
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This document was uploaded on 03/06/2014 for the course MATH 4740 at Cornell.
 Spring '10
 DURRETT
 The Land

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