Page 196 - A First Course In Stochastic Models

THE ERLANG DELAY MODEL 189

1 and so p 0 = 1 − λ/µ. Hence we find the explicit solution
i
p i = (1 − ρ)ρ , i = 0, 1, . . . (5.1.2)
with ρ = λ/µ. In particular, 1 − p 0 = ρ and so ρ can be interpreted as the long-
run fraction of time the server is busy. This explains why ρ is called the server
utilization. Let

L q = the long-run average number of customers in queue
(excluding any customer in service). The constant L q is given by

∞

L q = (j − 1)p j ,
j=1
as can be rigorously proved by assuming a cost at rate k whenever k customers are
waiting in queue and applying Theorem 4.2.2. Substituting (5.1.2) into the formula
for L q , we obtain
ρ 2
L q = ,
1 − ρ
in agreement with the Pollaczek–Khintchine formula for the general M/G/1 queue.
To determine the waiting-time probabilities we need the so-called customer-
average probabilities

π j = the long-run fraction of customers who find j other
customers present upon arrival, j = 0, 1, . . . .

In the M/M/1 case the customer-average probabilities π j are identical to the
time-average probabilities p j , that is,

π j = p j, j = 0, 1, . . . . (5.1.3)
This identity can be seen from the PASTA property. Alternatively, the identity can
be proved by noting that in a continuous-time Markov chain, p j q jk represents the
long-run average number of transitions from state j to state k ( = j) per time unit.
Thus in the M/M/1 case the long-run average number of transitions from state
j to state j + 1 per time unit equals λp j . In other words, the long-run average
number of arrivals per time unit finding j other customers present equals λp j .
Dividing λp j by the average arrival rate λ yields the customer-average probability
π j . The probability distribution {π j } is the equilibrium distribution of the embedded
Markov chain describing the number of customers present just before the arrival
epochs of customers. This probability distribution enables us to find the steady-state
waiting-time probabilities under the assumption of service in order of arrival. Let

W q (x) = lim P {D n ≤ x}, x ≥ 0, (5.1.4)
n→∞