Page 409 - A First Course In Stochastic Models

404 ALGORITHMIC ANALYSIS OF QUEUEING MODELS

this equation becomes

r 1 λ 1 r 2 λ 2
+ = η, (9.7.17)
cµ(1 − η) + λ 1 cµ(1 − η) + λ 2
where r 1 = 1 − bλ 1 /(λ 1 − λ 2 ) and r 2 = 1 − r 1 . Here it is assumed that λ 1 = λ 2 .
Once η is known, we can express p c2 into p c1 . Substituting p c+1,2 = ηp c2 into
(9.7.15) with n = c yields

(cµ + λ 2 )p c2 = cµηp c2 + λ 1 bp c1 .

The following algorithm can now be given.

Algorithm
Step 0. Calculate first η as the unique root of equation (9.7.17) on (0,1). Let p c1 := 1
and p c2 := λ 1 b{cµ(1 − η) + λ 2 } −1 p .
c1
Step 1. For k = c − 1, . . . , 0, use equation (9.7.14) with n = k + 1 and equation
(9.7.15) with n = k to solve for p k1 and p .
k2
Step 2. Calculate p := p +p n2 for n = 0, 1, . . . , c and next use relation (9.7.10)
n
n1
to normalize the p as
n
−1
 
c−1
p c
p n :=  p +  p , n = 0, 1, . . . , c.
n
j
1 − η
j=0
Generalized Erlangian arrivals
Suppose that the interarrival time has density
m i−1
i t −αt
a(t) = q i α e , t ≥ 0,
(i − 1)!
i=1
where q m > 0. In other words, with probability q i an interarrival time is the
sum of i independent phases each having an exponential distribution with mean
1/α. We again use the continuous-time Markov chain approach to compute the
probabilities p j . Define X(t) as the number of customers present at time t and let
Y(t) be the number of remaining phases of the interarrival time in progress at time
t. The process {(X(t), Y(t))} is a continuous-time Markov chain with state space
I = {(n, i) | n ≥ 0; 1 ≤ i ≤ m}. By equating the rate at which the system leaves
the set of states having at least n customers present to the rate at which the system
enters this set, we find

min(n, c)µp n = αp n−1,1 , n ≥ 1. (9.7.18)