This preview shows page 1. Sign up to view the full content.
Unformatted text preview: hat the queue is stable if the arrival rate is smaller than
the longrun service rate.
Theorem 3.5. Suppose < µ. If the queue starts with some ﬁnite number
k 1 customers who need service, then it will empty out with probability one.
Furthermore, the limiting fraction of time the server is busy is
/µ.
Proof. Let Tn = t1 + · · · + tn be the time of the nth arrival. The strong law of
large numbers, Theorem 3.2 implies that
1
Tn
!
n
Let Z0 be the sum of the service times of the customers in the system at time
0 and let si be the service time of the ith customer to arrive after time 0. The
strong law of large numbers implies
Z0 + Sn
1
!
n
µ
The amount of time the server has been busy up to time Tn is Z0 + Sn . Using
the two results
Z0 + Sn
!
Tn
µ
The actual time spent working in [0, Tn ] is Z0 + Sn Zn where Zn is the amount
of work in the system at time Tn , i.e., the amount of time needed to empty the
system if there were no more arrivals. To argue that equality holds we need
to show that Zn /n ! 0. Intuitively, the condition < µ implies the queue
reaches...
View
Full
Document
This document was uploaded on 03/06/2014 for the course MATH 4740 at Cornell University (Engineering School).
 Spring '10
 DURRETT
 The Land

Click to edit the document details