Page 423 - A First Course In Stochastic Models

418 ALGORITHMIC ANALYSIS OF QUEUEING MODELS

Table 9.8.4 The minimal buffer size in the M/G/c/c + N queue
ρ = 0.5 ρ = 0.8
α 10 −2 10 −4 10 −6 10 −8 10 −10 10 −2 10 −4 10 −6 10 −8 10 −10
c=1
2 1
c = exa 4 9 15 20 25 10 26 41 57 73
S 2
app 4 10 15 20 25 10 26 41 57 73
2
c = 2 exa 7 16 26 35 45 19 49 80 110 141
S
app 7 17 25 36 46 19 50 80 111 140
c=10
2 1
c = exa 1 7 12 17 23 8 24 39 55 71
S 2
app 1 7 13 17 23 8 24 39 55 71
2
c = 2 exa 1 10 20 29 39 14 44 74 105 135
S
app 1 10 20 29 39 14 45 75 106 135
assumed that the number of time slots needed to serve a packet has a geometric
distribution {(1 − r) n−1 r, n ≥ 1}. The case of deterministic services is included as
a special case (r = 1). In many telecommunication applications the service time
of a packet is deterministic and equals one time slot. A served packet leaves the
system at the end of the time slot in which the service is completed. The numbers of
packets arriving in the system during consecutive time slots are independent non-
negative random variables with the common probability distribution {a n , n ≥ 0}.
It is assumed that the packets arrive individually during the time slots and that
an arriving packet is rejected when it finds the buffer full upon arrival. It is no
restriction to use the convention of individual arrivals provided that the partial
rejection strategy applies when arrivals actually occur in batches. The load factor
ρ is defined as
λµ
ρ = ,
c
∞

where λ = na n is the arrival rate of new packets and µ = 1/r is the mean
n=1
service time of a packet. Let
P rej = the long-run fraction of packets that are rejected.
Under the assumption of ρ < 1 an approximation to P rej can be given in terms
of the state probabilities in the corresponding infinite-buffer model. Assuming that
(∞)
ρ < 1, define for the infinite-buffer model the probability u by
j
(∞)
u = the long-run fraction of time slots at whose beginnings there are
j
j packets in the system
for j = 0, 1, . . . . By the assumption of geometrically distributed service times, the
process describing the number of packets present at the beginning of a time slot is