Page 41 - A First Course In Stochastic Models

32 THE POISSON PROCESS AND RELATED PROCESSES

the types 1, . . . , L arrive according to independent non-stationary Poisson processes with
respective arrival rate functions p 1 λ(t), . . . , p L λ(t).
1.26 Consider the infinite-server queueing model from Section 1.1.3, but assume now that
customers arrive according to a non-stationary Poisson process with arrival rate function
λ(t). Let B(x) be the probability distribution function of the service time of a customer.
Assuming that the system is empty at epoch 0, prove that the number of busy servers at
t

time t has a Poisson distribution with mean 0 λ(x){1 − B(t − x)}dx.
1.27 Consider the M/G/∞ queue from Section 1.1.3 again. Let the random variable L be
the length of a busy period. A busy period begins when an arrival finds the system empty
and finishes when there are no longer any customers in the system. Argue that P{L > t}
can be obtained from the integral equation
t

−λx
P{L > t} = 1 − B(t) + {B(t) − B(x)}P{L > t − x}λe dx, t ≥ 0,
0
where B(t) is the probability distribution function of the service time of a customer. Remark:
it was shown in Shanbhag (1966) that the Laplace transform of P{L > t} is given by
−1
1 λ + s 1 ∞ x
1 − + exp −sx − λ (1 − B(y))dy dx .
s λ λ 0 0
BIBLIOGRAPHIC NOTES

A treatment of the Poisson process can be found in numerous texts. A good treat-
ment is given in the books of Ross (1996) and Wolff (1989). The Poisson process
is fundamental to all areas of applied probability. The infinite-server queue with
Poisson input has many applications. The applications in Examples 1.1.3 and 1.1.4
are taken from papers of Parikh (1977) and Sherbrooke (1968).

REFERENCES

Adelson, R.M. (1966) Compound Poisson distributions. Operat. Res. Quart. 17, 73–75.
Den Iseger, P.W., Smith, M.A.J. and Dekker, R. (1997) Computing compound Poisson
distributions faster. Insurance Mathematics and Economics, 20, 23–34.
Glaz, J. and Balakrishnan, N. (1999) Scan Statistics and Applications. Birkh¨ auser, Boston.
Khintchine, A.Y. (1969) Mathematical Methods in the Theory of Queueing. Hafter, New
York.
Parikh, S.C. (1977) On a fleet sizing and allocation problem. Management Sci., 23, 972–977.
Ross, S.M. (1996) Stochastic Processes, 2nd edn. John Wiley & Sons, Inc., New York.
Shanbhag, D.N. (1966) On infinite server queues with batch arrivals. J. Appl. Prob., 3,
274–279.
Sherbrooke, C.C. (1968) Metric: a multi-echelon technique for recoverable item control,
Operat. Res., 16, 122–141.
Wolff, R.W. (1989) Stochastic Modeling and the Theory of Queues. Prentice Hall, Englewood
Cliffs, NJ.