Queueing System G/M/1

Consider a single-server queueing system with exponentially distributed service times which are independent of each other and of the arrival process, and with an arrival process where the successive interarrival times are independent and identically distributed. We will let a denote the

parameter of the service time distribution and let φ denote the interarrival time distribution. Then q_n defined by (5.3) becomes the probability that the server completes exactly n services during an interarrival time provided that there are that many customers available to be served. We define r_n = q_{n + 1} + q_{n + 2} + · · · as before and let

so that r is the expected number of services which the server is capable of completing during an interarrival time.

If r ≥ 1, then the server can keep up with arrivals, and we expect the queue size to be recurrent; if r

< 1, then the queue size is likely to increase to infinity, and the process is transient. Note that this queueing system is the dual of the system M/G/1 studied in the preceding section. This duality comes about by the reversed roles of the interarrival times and service times. Therefore, with r fixed, when one system is transient the other is recurrent and vice versa. Later we will make these heuristic remarks more precise. The duality mentioned extends somewhat further and enables us to obtain the limiting distribution of the queue size for the G/M/1 system as a function of the probabilities of never becoming empty in the M/G/1 system. There is a similar functional relationship between the limiting distribution of the queue size for the M/G/1 system and the probabilities of never becoming empty for the G/M/1 system.

In this section we denote by the number of customers present in the system just before the time T_n of the nth arrival.

(6.1) THEOREM. is a Markov chain with state space E = {0, 1, . . .} and transition matrix

Proof. Let M_{n + 1} be the number of services completed during the (n + 1)th interarrival time [T_n, T_n

+ 1). Then

that is, is the number of customers that were there before the nth arrival plus the nth customer less those whose services were completed during the next interarrival time. It follows from (6.3) that to show that X* is a Markov chain, it is sufficient to show that M_{n + 1} is conditionally independent of

the past history before T_n given the present number . But this follows from the independence of the service times and interarrival times and the fact that by the memorylessness of the exponential distribution, the remaining service time of the customer who was being served at time T_n (if any) has the same exponential distribution as any service time.

Furthermore, the above reasoning shows that as long as there are customers to be served, the conditional distribution of the number of services completed will be Poisson: with Z = T_{n + 1} − T_n,

Taking expectations with respect to Z, which is independent of , we obtain

Putting (6.3) and (6.4) together, we see that the transition probabilities P*(i, j) are as claimed.

It is clear from an inspection of (6.2) that the Markov chain X* is irreducible aperiodic. Therefore all the states are of the same type. The following theorem shows that they are all recurrent non-null if and only if r > 1.

(6.5) THEOREM. X* is recurrent non-null if and only if r > 1. If r > 1, the limiting distribution

is given by

where β is the unique number satisfying

If r ≤ 1, then π*(j) = 0 for all j.

Proof. By Theorem (2.1), X* is recurrent non-null if and only if ν = νP*, ν1 = 1 has a solution; and if there is a solution, then π* = ν. Writing out the equations for ν = νP*, we obtain (observe that the initial equation uses r_n = q_{n + 1} + q_{n + 2} + · · ·)

Let a function f be defined by

Now the equation for ν₀ gives an equation for f(1), summing the equations for ν₀ and ν₁ yields an equation for f(2), and so on. We obtain

In other words, f satisfies

with Q as defined by (5.32), and we are interested in a solution of this satisfying

Remembering that Q was obtained from P by deleting the 0th row and column, we conclude by Proposition (4.5) that such a solution f exists if and only if the chain X is transient. By Theorem (5.34), this is true if and only if r > 1. If r > 1, the solution for f is given by (5.35), where β satisfies (5.36), which is the same relation as (6.8).

Solving for ν, by using (6.9), out of the expression (5.35) for f, we obtain

which is the same as the claimed limiting distribution π*. This completes the proof, since the last statement, that r ≤ 1 implies π* = 0, follows directly from Theorem (5.3.2).

The limiting distribution obtained in the recurrent non-null case, then, is a geometric distribution. It is easy to work with, and in particular, we see that

It is worth mentioning that this last result may be used, in practice, to estimate β directly; so after ascertaining that the model fits the system of practical interest, all one needs is an estimate of the average number of customers in the system just before an arrival. Once this number is obtained, (6.10) may be used to estimate by setting . In other words, if one is interested only in the queue-size process, one does not need to estimate the probabilities q_n or try to solve the equation (6.8) for β.

It is easy to relate these results to the waiting times of the individual customers. Let W_n be that of the nth customer. Then, if the service is on a first-come, first-served basis, the nth customer’s waiting time will be equal to the sum of the service times of those customers whom he found there upon his arrival. That is, W_n is equal to the sum of random variables each of which has the exponential distribution with parameter a. From the interpretation of the geometric distribution in its relation to the time of first success in a sequence of Bernoulli trials (cf. Chapter 3 for details), we may think of W_n for large n as the time of first success with trials being performed at the times of arrivals in a Poisson process with rate a (note that there is a trial at time 0 in this setup). From the results on decomposition of Poisson processes (cf. Chapter 4) we see that W_n will have an exponential distribution, for large n, with parameter equal to a times the probability of success 1 − β, except that at t = 0 there is a probability of immediate success equal to 1 − β. Hence,

Some further quantities of interest can be obtained from this by straight-forward methods. We leave some of these as exercises.

We return to the queue size process X* and consider it when it is not recurrent non-null, namely, when r ≤ 1. The next theorem shows that when r < 1, X* is transient and when r = 1, X* is recurrent null. In the transient case we compute the probability that the queue never becomes empty given the number of initial customers.

(6.12) THEOREM. X* is transient if and only if r < 1. If r < 1, the probability f*(j) that the queue starting with j customers never becomes empty is given by

where the π(j) are given by (5.21), (5.22), and (5.23).

Proof. In view of Proposition (4.5), f* is the solution of h = Q*h, 0 ≤ h ≤ 1, where Q* is obtained from P* by deleting the 0th row and column. Then the equations for h = Q*h become (write f*(j) = h_j)

Define π₀ = q₀h₁, π₁ = (1 − q₀)h₁, and let π_j = h_j − h_{j − 1} for j = 2, 3, . . . . Then the first equation of (6.14), along with π₀ = q₀h₁, implies the two equations

and subtracting the equation for h_{j − 1} from the one for h_j for j = 2, 3, . . . yields

In other words, π satisfies π = πP with P as given in (5.2), and we are interested in the solution of π = πP with ∑π_j = lim_j h_j = 1. From Theorem (5.14), such a solution exists if and only if r < 1. When r <

1, the solution is given by (5.21), (5.22), and (5.23), and the solution π is connected to h by the relation h_j = π₀ + · · · + π_j. This completes the proof.

A special case of some interest is the single-server queueing system with Poisson arrivals and exponential service times (usually denoted by M/M/1 queue). One may consider it either as a special M/G/1 queue or as a special G/M/1 queue. It is easier, from the computational point of view, to take it as a special case of the G/M/1 queue with the interarrival distribution

(and, of course, exponential service times with parameter a).

To compute the limiting distribution of the queue size just before the nth arrival, we use Theorem (6.5). Now, to solve for β in (6.8), first note that

Thus, equation (6.8) becomes

We know that 1 is a solution. To obtain the other, we see that when r = a/λ > 1 we have, as the smallest solution,

So we have

It turns out, though it is not at all apparent from anything which precedes this, that we also have

for the queue size Y_t at time t, and similarly

for the queue size X_n just after the nth departure (cf. Exercise (8.21) for this).

For a different formulation of this problem we refer the reader to Section 6 of Chapter 8. For the general G/M/1 queue and its time-dependent behavior, see Section 7 of Chapter 10.