The M/M/Infinity Service System with Ranked Servers in Heavy Traffic

We consider a stream of customers, with independent exponentially distributed interarrival times, arriving at rate λ to an infinite sequence S 1 , S 2 , . . . of servers. Each arriving customer engages the server S l having the lowest index among currently idle servers, and renders that server busy for an independent exponentially distributed service time with mean 1. This stochastic service system, which is conventionally denoted M/M/∞, has been extensively studied in the limit λ → ∞; see Newell [N]. We shall be interested in a question mentioned only tangentially by Newell: what is the distribution of the random variable L defined as the index of the server S L serving a newly arriving customer when the system is in equilibrium? Newell [N, p. 9] states that L "is approximately uniformly distributed over the interval" [1, λ], basing this assertion on the approximation

(1.1)

But no error bounds are given for this or other approximations stated by Newell, and not even the fact that the first moment has the asymptotic behavior

that it would have under the uniform distribution is established rigorously. Our goal in this paper is to give a rigorous version of (1.1) that will suffice to establish not only (1.2), but also the next term,

and more generally

for m ≥ 1. In particular, we have Var

Since the interval [0, 1] is bounded, formula (1.4) shows that the m-th moment of L/λ tends to 1/(m + 1) as λ → ∞ for all m ≥ 1, and thus suffices to show that the distribution of L/λ tends to the uniform distribution on the interval [0, 1]. We note that a problem that is in a sense dual to ours (finding the largest index of a busy server, rather than the smallest index of an idle server) has been treated by Coffman, Kadota and

The key to our results is the probability Pr[L > l], which is simply the probability that the first l servers S 1 , . . . , S l are all busy. It is well known that this probability is given by the Erlang loss formula

where

(1.5) (see for example Newell [N, p. 3]). The sum D l can be expressed as an integral,

(see for example Newell [N, p. 7]), and most of Newell’s analysis is based on such a representation. But we shall work directly with the expression of D l as the sum in (1.5).

We shall divide the range of summation in (1.5) into two parts. The first, which we shall call the “body” of the distribution, will be 0

The second, which we shall call the “tail”, will be l > l 0 . In Section 2, we shall derive an estimate for Pr [L > l] in the body, and in Section 3, we shall derive an estimate for the tail. In Section 4, we shall combine these estimates to establish (1.4).

In this section we shall establish the estimate

for l ≤ l 0 = λ -s, where s = √ λ. We begin by using the principle of inclusion-exclusion to derive bounds on the denominator D l .

We begin with a lower bound. Since

we have

.

For the first sum we have

We note that the logarithm of (l/λ) l has a non-negative second derivative for l ≥ 1. Thus (l/λ) l assumes its maximum in the interval 0 ≤ l ≤ l 0 for l = 0, l = 1 or l = l 0 . Its values there are 0, 1/λ and (1

, respectively. As λ → ∞, the largest of these values is 1/λ, so

For the second sum we have

The logarithm of l 2 (l/λ) l has a non-negative second derivative for l ≥ 3, so an argument similar to that used for the first sum shows that O l 2 (l/λ) l = O(1/λ) for 0 ≤ l ≤ l 0 . Thus we have

and the lower bound

For an upper bound, we have

.

For the third sum we have

and thus the upper bound

Combining this upper bound with the lower bound (2.2) yields

To obtain Pr[L > l], we take the reciprocal of D l :

, we obtain (2.1).

In this section we shall establish the estimate

for l ≥ λ -s, where s = √ λ. To obtain an upper bound on Pr[L > l], we obtain a lower bound on D l . We have

because l -⌊λ -s⌋ ≥ l -(λ -s) ≥ 0 by assumption and ⌊λ -2s⌋ ≥ 0 for all sufficiently large λ. There are ⌊λ -2s⌋ -⌊λ -2s⌋ + 1 ≥ s terms in the sum (3.2). Furthermore, the smallest of these terms is the last, because its denominator contains factors of λ where the preceding terms contain factors smaller than λ. Thus we have

For the factorial in the denominator of this bound, we shall use the estimate n! ≤ e

√ n e -n n n , which holds for all n ≥ 1 (because the trapezoidal rule underestimates the integral n 1 log x dx of the concave function log x). This estimate yields

e ⌊λ -2s⌋ ⌊λ -2s⌋ ⌊λ-2s⌋ λ l-⌊λ-2s⌋ .

We have e ⌊λ-2s⌋ ≥ e λ-2s-1 ,

and ⌊λ -2s⌋ ≤ s.

Substituting these bounds into (3.3) yields

Taking the reciprocal of this bound yields (3.1).

In this section we shall use (2.1) and (3.1) to prove (1.4). We write

for the backward differences of the m-th powers of l. Then partial summation yields

This formula shows that we should evaluate sums of the form

We shall show that

Substitution of this formula into (4.1) will then yield (1.4).

We shall break the range of summation in (4.2) at l 0 = λ -s, where s = √ λ, using (2.1) for 0 ≤ l ≤ l 0 and (3.1) for l > l 0 . Summing the

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