zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
On a batch arrival Poisson queue with a random setup time and vacation period. (English) Zbl 1040.90511
Summary: This paper deals with the steady state behaviour of a single server batch arrival Poisson queue with a random setup time and a vacation period. The service of the first customer in each busy period is preceded by a random setup period, on completion of which service starts. As soon as the system becomes empty the server goes on vacation for a random length of time. On return from vacation, if he finds customer(s) waiting, the server starts servicing the first customer in the queue. Otherwise it takes another vacation and so on. We study the steady state behaviour of the queue size distribution at random (stationary) point of time as well as at departure point of time and try to show that departure point queue size distribution can be decomposed into three independent random variables, one of which is the queue size of the standard $M^x/M/1$ queue. The interpretation of the other two random variables will also be provided. Further, we derive analytically explicit expressions for the system state (number of customers in the system) probabilities and provide their appropriate interpretations. Also, we derive some system performance measures. Finally, we develop a procedure to find mean waiting time of an arbitrary customer.

MSC:
90B22Queues and service (optimization)
60K25Queueing theory
WorldCat.org
Full Text: DOI
References:
[1] Boxima, O. J.; Groenendijk, W. P.: Waiting time in discrete time cyclics service systems. IEEE trans. Commun. 36, 164-170 (1988) · Zbl 0655.90026
[2] Bruke, P. J.: Delays in single server queues with batch input. Ops. res. 23, 830-833 (1975)
[3] Chaudhry, M. L. and Templeton, J. G. C., A First Course in Bulk Queues. Wiley, New York, 1983 · Zbl 0559.60073
[4] Choudhury, G.: On a Poisson queue with general setup time and vacation period. Indian J. Pure appl. Math. 27, No. 12, 1199-1211 (1996) · Zbl 0862.60082
[5] Doshi, B. T.: A note on stochastic decomposition in GI/G/1 queue with vacations or setup time. J. appl. Prob. 22, 419-428 (1985) · Zbl 0566.60090
[6] Doshi, B. T.: Single server queues with vacations -- a survey. Queueing systems 1, 29-66 (1986) · Zbl 0655.60089
[7] Fuhrmann, S. W.; Cooper, R. B.: Stochastic decompositions in the M/G/1 queue with generalized vacation. Ops. res. 33, 1117-1129 (1985) · Zbl 0585.90033
[8] Gaver, D. P.: Imbedded Markov chain analysis of a waiting time process in continuous time. Ann. math. Stat. 30, 698-720 (1959) · Zbl 0087.33601
[9] Kleinrock, L., Queueing Systems, Vol. 1. Wiley, New York, 1975 · Zbl 0334.60045
[10] Lee, H. W.; Lee, S. S.; Park, J. O.; Chae, K. C.: Analysis of the mx/G/1 queue with N-policy and multiple vacations. J. appl. Prob. 31, 476-496 (1994) · Zbl 0804.60081
[11] Lee, S. S.; Lee, H. W.; Yoon, S. H.; Chae, K. C.: Batch arrival queue with N-policy and single vacation. Comput. ops. Res. 22, 173-189 (1995) · Zbl 0821.90048
[12] Levy, H.; Kleinrock, L.: A queue with starter and a queue with vacations. Ops. res. 34, 426-436 (1986) · Zbl 0611.60092
[13] Medhi, J., Stochastic Models in Queueing Theory. Academic Press, San Diego, 1991 · Zbl 0743.60100
[14] Medhi, J.: Extensions and generalizations of the classical single server queueing system with Poisson input. J. ass. Sci. soc. 36, 35-41 (1994)
[15] Medhi, J.; Templeton, J. G. C.: A Poisson input queue under N-policy with a general startup time. Comput. ops. Res. 34, 35-41 (1992) · Zbl 0742.60100
[16] Miller, L.: A note on busy period of an M/G/1 finite queue. Ops. res. 23, 1179-1182 (1975) · Zbl 0339.60082
[17] Sharma, O. P., Markovian Queues. Ellis Horwood, Chichester, 1990
[18] Takagi, H., Queueing Analysis: A Foundation of Performance evaluation, Vol. 1, Vacation and Priority Systems, Part. 1. North Holland, Amsterdam, 1991 · Zbl 0744.60114
[19] Lee, H. S.; Srinivasan, M. M.: Control policies for mx/G/1 queueing system. Mgmt. sci. 35, 708-721 (1989) · Zbl 0671.60099
[20] Wolff, R. W.: Poisson arrivals see time average. Ops. res. 30, 223-231 (1982) · Zbl 0489.60096
[21] Wolff, R. W., Stochastic Modeling and The Theory of Queues. Prentice-Hall International, Inc., London, 1989 · Zbl 0701.60083