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Unformatted text preview: y says that if one can be found, the other can also be found. A(τ ) ✛ ✛ W1
0 ✛
✲ W2 W3 ♣♣ ♣♣ ♣♣ ♣♣ ♣♣ ♣♣ ♣♣ ♣♣
♣♣♣♣♣♣♣♣ ✲ ✲ D(τ )
S1 t S2 Figure 3.13: Arrival process, departure process, and waiting times for a queue. Renewals occur at S1 and S2 , i.e., when an arrival sees an empty system. The area
Rt
between A(τ ) and D(τ ) up to time t is 0 L(τ ) dτ where L(τ ) = A(τ ) − D(τ ). The sum
W1 + · · · + WA(t) also includes the shaded area to the right of t. Figure 3.13 illustrates the setting for Little’s theorem. It is assumed that an arrival occurs
at time 0, and that the subsequent interarrival intervals are IID. A(t) is the number of
arrivals from time 0 to t, including the arrival at 0, so {A(t) − 1; t ≥ 0} is a renewal
counting process. The departure process {D(t); t ≥ 0} is the number of departures from
0 to t, and thus increases by one each time a customer leaves the system. The diﬀerence,
L(t) = A(t) − D(t), is the number in the system at time t.
To be speciﬁc about the waiting time of each customer, we assume that customers enter
service in the order of their arrival to the system. This service rule is called FirstComeFirstServe (FCFS).9 Assuming FCFS, the system time of customer n, i.e., the time customer
n spends in the system, is the interval from the nth arrival to the nth departure. Finally,
9 For single server queues, this is also frequently referred to as First In First Out (FIFO) service. 118 CHAPTER 3. RENEWAL PROCESSES the ﬁgure shows the renewal points S1 , S2 , . . . at which arriving customers ﬁnd an empty
system. As observed in Example 3.1.1, the system probabilistically restarts at each of these
renewal instants, and the behavior of the system in one interrenewal interval is independent
of that in each other interrenewal interval.
It is important here to distinguish between two diﬀerent renewal processes. The arrival
process, or more precisely, {A(t) − 1; t ≥ 0} is one renewal counting process, and the
renewal epochs S1 , S2 , . . . in the ﬁgure generate another renewal process. In what follows,
{A(t); t ≥ 0} is referred to as the arrival process and {N (t); t ≥ 0}, with renewal epochs
S1 , S2 , . . . is referred to as the renewal process. The entire system can be viewed as starting
anew at each renewal epoch, but not at each arrival epoch.
We now regard L(t), the number of customers in the system at time t, as a reward function
over the renewal process. This is slightly more general than the reward functions of Sections
3.4 and 3.5, since L(t) depends on the arrivals and departures within a busy period (i.e.,
within an interrenewal interval). Conditional on the age Z (t) and duration X (t) of the interrenewal interval at time t, one could, in principle, calculate the expected value R(Z (t), X (t))
over the parameters other than Z (t) and X (t). Fortunately, this is not necessary and we
can use the sample functions of the combined arrival and departure processes directly,
which specify L(t) as A(t) − D(t). Assuming that the expected interrenewal interval is
ﬁnite, Theorem 3.6 asserts that the time average number of customers in the system (with
probability 1) is equal to E [Ln ] /E [X ]. E [Ln ] is the expected area between A(t) and D(t)
(i.e., the expected integral of L(t)) over an interrenewal interval. An interrenewal interval
is a busy period followed by an idle period, so E [Ln ] is also the expected area over a busy
period. E [X ] is the mean interrenewal interval.
From Figure 3.13, we observe that W1 + W2 + W3 is the area of the region between A(t)
and D(t) in the ﬁrst interrenewal interval for the particular sample path in the ﬁgure. This
is the aggregate reward over the ﬁrst interrenewal interval for the reward function L(t).
More generally, for any time t, W1 + W2 + · · · + WA(t) is the area between A(t) and D(t)
Rt
up to a height of A(t). It is equal to 0 L(τ )dτ plus the remaining waiting time of each of
the customers in the system at time t (see Figure 3.13). Since this remaining waiting time
is at most the area between A(t) and D(t) from t until the next time when the system is
empty, we have
N (t) X n=1 Ln ≤ Z A(t) t τ =0 L(τ ) dτ ≤ X
i=1 N (t)+1 Wi ≤ X Ln . (3.56) n=1 Assuming that the expected interrenewal interval, E [X ], is ﬁnite, we can divide both sides
of (3.56) by t and go to the limit t → 1. From the same argument as in Theorem 3.6, we
get
lim t→1 PA(t)
i=1 t Wi = lim t→1 Rt τ =0 L(τ ) dτ t = E [Ln ]
E [X ] with probability 1. (3.57) Rt
We denote limt→1 (1/t) 0 L(τ )dτ as L . The quantity on the left of (3.57) can now be
broken up as waiting time per customer multiplied by number of customers per unit time, 3.6. APPLICATIONS OF RENEWALREWARD THEORY 119 i.e.,
lim t→1 PA(t)
i=1 t Wi = lim t→1 PA(t) Wi
A(t)
lim
.
t→1
A(t)
t i=1 (3.58) From (3.57), the limit on the left side of (3.58) exists (and equals L) with probability 1.
The second limit on the right also exists with probabi...
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This note was uploaded on 09/27/2010 for the course EE 229 taught by Professor R.srikant during the Spring '09 term at University of Illinois, Urbana Champaign.
 Spring '09
 R.Srikant

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