4.4 SOME IMPORTANT RELATIONSHIPS IN QUEUEING THEORY
We mentioned in Section 4.1 that queueing theory has been highly successful in deriving expressions for the low moments of quantities such as the waiting times of the users of a queueing system or the number of users present in the system at a given time. We shall now begin our discussion with the derivation of a few important relationships involving the expected values of these quantities, , q, , and q for a very general queueing system. J. D. C. Little [LITT 611 is generally credited with being the first to prove these relationships formally. Subsequent authors have shown that Little's results are valid for queueing systems more general than he assumed in his original work [STID 74]. Here we shall use an informal and intuitive argument paralleling the one offered by Kleinrock [KLEI 75].
The search for the relationships is motivated by the intuition-satisfying notion that the average length of the queue in front of a facility offers a good indication of the average waiting time for use of that facility (and vice versa). These relationships turn out to be especially simple.
First, we shall position ourselves at the "entrance" of a queueing system and count the number of users that arrive there during an interval of arbitrary length, , beginning at the time t = 0 when the system is empty (see also Figure 4.3). We let
Next, we count the number of users leaving the system at its "exit" and let
If the system is empty at t = 0, the number of users in the system at the time t = is given by
We can now use (4.6) to express the total amount of time, l(), spent by all users in the queueing system during the interval [0,l. We have
Clearly, 1() represents the area between the functions a() and c(), as illustrated in Figure 4.3.
The average number of users () in the queueing system during the interval [0, ] can now be obtained by dividing the total amount of time spent by all users in the queueing system, l(), by the time :
We have written (4.8) in this form because both ratios on its right-hand side have very real physical meaning. The ratio a()/ is simply the average number of arrivals per unit of time (the arrival rate) during the interval and can be indicated, given our earlier notation, as . Similarly, l()/a() is the average time spent by a user in the queueing system during the interval [0, ] and can be indicated, given our earlier notation, as . We can then write
() = (4.9)
If we now let the length of the interval, , tend to infinity, it is clear from our earlier definitions of , , and that these quantities represent the limits of () , and respectively. So if the limits of the last two quantities ( , and ) actually exist, the limit of () also exists and from (4.9) we have the relationship
This is one of the best-known results of queueing theory and is referred to as Little's formula. Later in this chapter we shall explore the conditions under which the limits of and of () exist for many types of queueing systems. A few important remarks are in order:
In deriving (4.10), we stationed ourselves at the "entrance" and "exits" of the queueing system. We could have performed exactly the same analysis if we had counted entries to the queueing system (as before) but exits from the queue (or, in other words, entries to the service facility). In that case we would have derived the result
q = q (4.11)
where q and q are the average number and average stay in the queue, as defined earlier.
In a similar way, but by focusing now on the service facility itself (i.e., by counting entries and exits from the service facility), we could show that
where s, is the (steady-state) average number of users in the service facility. [Note that this result is independent of the number of servers, m. What matters on the right-hand side of (4.12) is the average amount of time a user spends in the facility, E[S].]In our discussion so far, we have never specified a queueing discipline. Thus, (4. 10) and (4.11) hold irrespective of the method used to determine the order of entry to the service facility. [See also Section 4.9.] They also hold for the case where users belong to a number of distinct classes to which difrerent levels of priority are assigned. Within each of the classes, (4. 10) and (4.11) are valid. The only condition that was placed on the arrival process at the queueing system is that the quantity a()/ , the long-term arrival rate, be finite. To appreciate the significance of this, consider a case in which the arrival rate, rather than being a constant, is a function of some parameter-say, of the total number of users present in the queueing system or of time. Then, we can still use (4.10) and (4.11), with a value of equal to the long-term average of the rate at which users enter the system. Similarly, for queueing systems with finite queue capacity, for which there is the possibility that some potential users will be turned away, we use (4. 10) and (4.11) with A equal to the average rate at which users actually join the queueing system. We shall see several examples of this type in subsequent sections.
A last relationship of importance which is always valid due to (4.1) is
Note that (4.10), (4.11), and (4.13) make it possible, with given and , to compute all four of the quantities , q, , and q if any one of them can be determined.
Finally, it is worth remembering the following convenient argument (not "proof") that leads to (4.10) and (4.11). In the steady state, the average number of users that a random user finds in a FCFS "system" upon arrival should be equal to the number of users he or she leaves behind upon departure, with both of these numbers equal to (or q, depending on what the "system" is). But the average number of users left behind is simply the arrival rate times the average time a random user stays in the "system," (or q).
3 See also Section 4.9.