Page 425 - A First Course In Stochastic Models

420 ALGORITHMIC ANALYSIS OF QUEUEING MODELS

Table 9.8.5 Numerical results for the discrete-time queue
c = 1 c = 2
N Poisson Geometric N Poisson Geometric
1 exa 3.406×10 −1 4.737×10 −1 2 exa 2.379×10 −1 4.133×10 −1
app 3.024×10 −1 4.119×10 −1 app 1.879×10 −1 3.481×10 −1
5 exa 5.505×10 −2 1.260×10 −1 5 exa 6.595×10 −2 1.970×10 −1
app 5.504×10 −2 1.254×10 −1 app 6.044×10 −2 1.859×10 −1
10 exa 1.481×10 −2 5.081×10 −2 10 exa 1.693×10 −2 9.054×10 −2
app 1.481×10 −2 5.081×10 −2 app 1.592×10 −2 8.849×10 −2
50 exa 3.294×10 −6 5.178×10 −4 50 exa 3.702×10 −6 3.036×10 −3
app 3.294×10 −6 5.178×10 −4 app 3.511×10 −6 3.001×10 −3
100 exa 1.046×10 −10 2.656×10 −6 100 exa 6.626×10 −13 1.476×10 −5
app 1.046×10 −10 2.656×10 −6 app 6.283×10 −13 1.460×10 −5

distribution and the geometric distribution are considered for the distribution {a n }
of the number of arrivals during one time slot. In all examples we take the load
factor ρ = 0.9.
To conclude this section, it is noted that the approximation to P rej can be
extended to discrete-time queueing systems with correlated input. In many applica-
tions the input is not renewal but correlated. The switched-batch Bernoulli process
is often used for modelling correlated input processes. In this model there is an
underlying phase process that is alternately in the states 1 and 2, where the sojourn
times in the successive states are independent random variables that have a discrete
geometric distribution. The mean of the geometric sojourn time and the distribution
of the number of arrivals in a time slot depend on the state of the phase process.
Exercise 9.16 is to work out the approximation to P rej in this useful model with
correlated input.

EXERCISES

9.1 Consider the M/G/1 queue with exceptional first service. This model differs from the
standard M/G/1 queue only in the service times of the customers reactivating the server
after an idle period. Those customers have special service times with distribution function
B 0 (t), while the other customers have ordinary service times with distribution function B(t).
Use the regenerative approach to verify that the state probabilities can be computed from the
recursion scheme (9.2.1) in which λp 0 a j−1 is replaced by λp 0 a j−1 , where a n is obtained
by replacing B(t) by B 0 (t) in the integral representation for a n . Also, argue that p 0 satisfies
1−p 0 = λ[p 0 µ 0 +(1−p 0 )µ 1 ], where µ 1 and µ 0 denote the means of the ordinary service
times and the special service times.
9.2 Consider the M/G/1 queue with server vacations. In this variant of the M/G/1 queue
a server vacation begins when the server becomes idle. During a server vacation the server
performs other work and is not available for providing service. The length V of a server
vacation has a general probability distribution function V (x) with density v(x). If upon return
from a vacation the server finds the system empty, a new vacation period begins, otherwise