Page 400 - A First Course In Stochastic Models

MULTI-SERVER QUEUES WITH POISSON INPUT 395

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The M /D/c queue
Suppose that the service time of each customer is a constant D. Denoting by
p j (t) the probability that j customers are present at time t, we find by the same
arguments as used in Section 9.6.2 that

c c+j

p j (t + D) = p k (t)r j (D) + p k (t)r j−k+c (D), j = 0, 1, . . . ,
k=0 k=c+1
where the compound Poisson probability r j (D) is defined by

r j (D) = the probability that exactly j customers arrive during
a given time interval of length D, j = 0, 1, . . . .
Letting t → ∞, we find the system of linear equations

c c+j

p j = r j (D) p k + r j−k+c (D)p k , j = 0, 1, . . . (9.6.40)
k=0 k=c+1

together with the normalizing equation ∞ p j = 1. Just as in the M/D/c case,
j=0
this infinite system of equations can be reduced to a finite system of linear equations
by using the geometric tail behaviour of the p j . It holds that
p j ∼ στ −j as j → ∞, (9.6.41)

where τ is the unique root of the equation

c λD{1−β(τ)}
τ e = 1 (9.6.42)
on the interval (1, R) and the constant σ is given by
c−1
c
j
σ = [c − λDτβ (τ)] −1 p j (τ − τ ). (9.6.43)
′
j=0
∞ j
β
As before, β(z) = j=1 j z and the number R denotes the convergence radius
of the power series β(z). It is assumed that R > 1.
In general, however, it is computationally simpler to compute the state proba-
bilities p j by applying the discrete FFT method to the generating function P (z) =
∞ j
p
j=0 j z . In the same way as (9.6.6) was derived, we obtain
c−1
j c
p j (z − z )
j=0
P (z) = , (9.6.44)
1 − z e
c λD{1−β(z)}