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Unformatted text preview: lity 1 by the strong law for renewal
processes, applied to {A(t) − 1; t ≥ 0}. This limit is called the arrival rate ∏, and is equal
to the reciprocal of the mean interarrival interval for {A(t)}. Since these two limits exist
with probability 1, the ﬁrst limit on the right, which is the samplepathaverage waiting
time per customer, denoted W , also exists with probability 1. We have thus proved Little’s
theorem,
Theorem 3.8 (Little). For a FCFS G/G/1 queue in which the expected interrenewal interval is ﬁnite, the timeaverage number of customers in the system is equal, with probability
1, to the samplepathaverage waiting time per customer multiplied by the customer arrival
rate, i.e., L = ∏W .
The mathematics we have brought to bear here is quite formidable considering the simplicity
of the idea. At any time t within an idle period, the sum of customer waiting periods up to
time t is precisely equal to t times the timeaverage number of customers in the system up
to t (see Figure 3.13). Renewal theory informs us that the limits exist and that the edge
eﬀects (i.e., the customers in the system at an arbitrary time t) do not have any eﬀect in
the limit.
Recall that we assumed earlier that customers departed from the queue in the same order
in which they arrived. From Figure 3.14, however, it is clear that FCFS order is not
required for the argument. Thus the theorem generalizes to systems with multiple servers
and arbitrary service disciplines in which customers do not follow FCFS order. In fact, all
that the argument requires is that the system has renewals (which are IID by deﬁnition of
a renewal) and that the interrenewal interval is ﬁnite with probability 1. A(τ ) ✛ ✛
✛
0 W1 W2 ✲ W3 ♣♣
♣♣
♣♣ ✲ ♣♣
♣♣
♣♣ ♣♣
♣♣ ♣ ♣ ♣ ♣ ♣
♣♣ ♣♣ ♣♣ ♣♣ ♣♣ ♣♣ ✲
S1 t Figure 3.14: Arrivals and departures in nonFCFS systems. The aggregate reward
(integral of number of customers in system) up to time t is the enclosed area to the left
of t; the sum of waits of customers arriving by t includes the additional shaded area to
the right of t. Finally, suppose the interrenewal distribution is nonarithmetic; this occurs if the interarrival distribution is nonarithmetic. Then L, the timeaverage number of customers in the 120 CHAPTER 3. RENEWAL PROCESSES system, is also equal to10 limt→1 E [L(t)]. It is also possible (see Exercise 3.30) to replace
the timeaverage waiting time W with limn→1 E [Wn ]. This gives us the following variant
of Little’s theorem:
lim E [L(t)] = ∏W = lim ∏E [Wn ] .
n→1 t→1 (3.59) The same argument as in Little’s theorem can be used to relate the average number of
customers in the queue (not counting service) to the average wait in the queue (not counting
service). Renewals still occur on arrivals to an empty system, and the integral of customers
in queue over a busy period is still equal to the sum of the queue waiting times. Let Lq (t) be
Rt
the number in the queue at time t and let Lq = limt→1 (1/t) 0 Lq (τ )dτ be the timeaverage
queue wait. Letting W q be the timeaverage waiting time in queue,
Lq = ∏W q . (3.60) If the interrenewal distribution is nonarithmetic, then
lim E [Lq (t)] = ∏W q . t→1 (3.61) The same argument can also be applied to the service facility. The timeaverage of the
number of customers in the server is just the fraction of time that the server is busy.
Denoting this fraction by ρ and the expected service time by Z , we get
ρ = ∏Z . 3.6.2 (3.62) Expected queueing time for an M/G/1 queue For our last example of the use of renewalreward processes, we consider the expected
queueing time in an M/G/1 queue. We again assume that an arrival to an empty system
occurs at time 0 and renewals occur on subsequent arrivals to an empty system. At any
given time t, let Lq (t) be the number of customers in the queue (not counting the customer
in service, if any) and let R(t) be the residual life of the customer in service. If no customer
is in service, R(t) = 0, and otherwise R(t) is the remaining time until the current service
will be completed. Let U (t) be the waiting time in queue that would be experienced by
a customer arriving at time t. This is often called the unﬁnished work in the queueing
literature and represents the delay until all the customers currently in the system complete
service. Thus the rv U (t) is equal to R(t), the residual life of the customer in service, plus
the service times of each of the Lq (t) customers currently waiting in the queue.
Lq (t) U (t) = X Zi + R(t). (3.63) i=1 10 To show this mathematically requires a little care. One approach is to split the reward function into
many individual terms. Let Ln (t) = 1 if the nth arrival since the beginning of the busy period has arrived
by time t, and let Ln (t) = 0 otherwise. Let Sn (t) = 1 if the nth departure since the beginning of the busy
period occurs by time t. It is easy to show that P direct Riemann integrability condition holds for each of
the
P
these reward functio...
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 Spring '09
 R.Srikant

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