Page 432 - A First Course In Stochastic Models

EXERCISES 427

(∞)
(b) For the infinite-buffer model, let π be the long-run fraction of packets that find j
j
other packets in the system upon arrival. Argue that
j 2 ∞
(∞) 1 (∞) (i)
π = u a s , j = 0, 1, . . . .
j λ k,i
k=0 i=1 s=j−k+1
(c) Develop a computer program for the discrete-time SBBP/D/c/c + N queue. Check
your computer program with the results below for the parameter values c = 3, ω 1 =
0.4, ω 2 = 0.2, α 1 = 1.4 and α 2 = 2.0. In case 1 a Poisson distribution is taken for each of
(1) (2) (1)
the distributions {a n } and {a n }; in case 2 a geometric distribution is taken for {a n } and
(2)
a Poisson distribution for {a n }.
N = 5 N = 10 N = 20 N = 30
Case 1 exa 1.683 × 10 −2 2.194 × 10 −4 3.908 × 10 −8 6.965 × 10 −12
app 1.085 × 10 −2 1.689 × 10 −4 3.024 × 10 −8 5.390 × 10 −12
Case 2 exa 3.781 × 10 −2 2.927 × 10 −3 3.101 × 10 −5 3.413 × 10 −7
app 2.603 × 10 −2 2.245 × 10 −3 2.506 × 10 −5 2.816 × 10 −7

9.17 Consider the D/M/c/c+N queue and the M/M/c/c+N queue with the same average
arrival rate and the same mean service time. For these two models, denote by N det (α) and
N exp (α) the smallest value of N for which the rejection probability is below a prespecified
1
level α. Verify experimentally that N det (α) ≈ N exp (α).
2
9.18 Consider the finite-capacity variants of the M/G/1 queue with exceptional first ser-
vice from Exercise 9.1, the M/G/1 queue with server vacations from Exercise 9.2 and the
M/G/1 queue with variable service rate from Exercise 9.3. Verify that the structural form
(9.8.4) for P rej remains valid for these queueing models. Do the same for the finite-capacity
variant of the M/M/c queue with impatient customers from Exercise 5.3.
X
9.19 Consider the batch-arrival M /G/c/N + c queue with complete rejection of a batch
when an arriving batch of customers does not find enough room in the buffer for the whole
batch. Let P rej denote the long-run fraction of customers who are rejected.
(a) Argue that
N+c
1
P rej = p k sβ s .
β
k=0 s>N+c−k
(∞)
(b) Using the approximation assumption p j ≈ γp for j = 0, 1, . . . , N +c−1, modify
j
the proof of part (b) of Theorem 9.8.2 to obtain the approximation (9.8.13) to P rej .
X
9.20 Consider the batch-arrival M /G/c/c+N queue with complete rejection. Suppose that
Q
the batch-size distribution {β j } has the property that β s = 1 for some 1 ≤ Q ≤ N +1.

s=1
(∞) X
Prove that p j = γp for 0 ≤ j ≤ N + c − Q for both the M /G/1/N + 1 queue and
j
X
the M /M/c/c + N queue. (Hint: define T, T j , T (∞) , T (∞) as in the proof of part
j
(∞)
(a) of Theorem 9.8.2 and let N k and N denote the number of service completions in
k
one cycle at which k customers are left behind. Argue first that E(N k ) = E(N (∞) ) for
k
(∞)
0 ≤ k ≤ N + c − Q. Next conclude that E(T j ) = E(T ) for 0 ≤ j ≤ N + c − Q, since
j
E(N j ) = λE[T j + . . . + T j+1−Q ] for 0 ≤ j ≤ N + c − Q.)