Page 424 - A First Course In Stochastic Models

FINITE-CAPACITY QUEUES 419

a discrete-time Markov chain. This Markov chain was analysed in Example 3.4.1
for the particular case of deterministic services. Letting

∞
(∞) j
U(z) = u z , |z| ≤ 1,
j
j=0
a minor modification of the Markov-chain analysis in Example 3.4.1 yields
c−1
(∞) c k k c
A(z) u [z (r + (1 − r)z) − z (r + (1 − r)z) ]
k
k=0
U(z) = c c , (9.8.15)
z − (r + (1 − r)z) A(z)
∞ n
a
where A(z) = n=0 n z . This expression is well suited for numerical purposes.
(∞) (∞)
First the c unknowns u , . . . , u are determined by computing the complex
0 c−1
roots of the denominator of (9.8.15); see Appendix G. Next the discrete FFT method
(∞)
can be applied to obtain the numerical values of the state probabilities u . In
j
order to obtain the approximation to P rej in the finite-buffer model, we need the
tail probability ∞ π (∞) for the infinite-buffer model. In the infinite-buffer
j=N+c j
(∞)
model the probability π is defined as
j
(∞)
π = the long-run fraction of packets who find j other packets
j
present upon arrival.
By the same arguments as used to obtain (9.8.10), we find
j ∞
(∞) 1 (∞)
π = u a s , j = 0, 1, . . . . (9.8.16)
j k
λ
k=0 s=j−k+1
The proposed approximation to P rej in the finite-buffer model is
∞
(∞)
(1 − ρ) π
j
j=N+c
P rej ≈ . (9.8.17)
∞
(∞)

1 − ρ π
j
j=N+c
It has been shown in Gouweleeuw and Tijms (1998) that for the single-server
case this approximation is asymptotically exact for large N (more precisely, the
approximation (9.8.17) is exact for the single-server case when the probability of
more than N arrivals during one time slot equals zero). In general it turns out
that (9.8.17) provides an excellent approximation to the rejection probability. To
illustrate this, Table 9.8.5 gives some numerical results for the case of determin-
istic service times. The number of servers is c = 1 and c = 2, while the Poisson