## A note on $$M/G/1$$ vacation systems with sojourn time limits.(English)Zbl 1255.90055

Summary: We deal with an $$M/G/1$$ vacation system with the sojourn time (wait plus service) limit and two typical vacation rules, i.e. multiple and single vacation rules. Using the level crossing approach, we derive recursive equations for the steady-state distributions of the virtual waiting times in $$M/G/1$$ vacation systems with a general vacation time and two vacation rules.

### MSC:

 90B25 Reliability, availability, maintenance, inspection in operations research 60K25 Queueing theory (aspects of probability theory) 60G17 Sample path properties 45D05 Volterra integral equations
Full Text:

### References:

 [1] Asmussen, S. (2003). Applied Probability and Queues (Appl. Math. 51 ), 2nd edn. Springer, New York. · Zbl 1029.60001 [2] Brill, P. H. (2008). Level Crossing Methods in Stochastic Models (Internat. Ser. Operat. Res. Manag. Sci. 123 ). Springer, New York. · Zbl 1157.60003 [3] Fuhrmann, S. W. and Cooper, R. B. (1985). Stochastic decompositions in the M/G/1 queue with generalized vacations. Operat. Res. 33, 1117-1129. · Zbl 0585.90033 [4] Gavish, B. and Schweitzer, P. J. (1977). The Markovian queue with bounded waiting time. Manag. Sci. 23, 1349-1357. · Zbl 0372.60134 [5] Hokstad, P. (1979). A single-server queue with constant service time and restricted accessibility. Manag. Sci. 25, 205-208. · Zbl 0414.60087 [6] Katayama, T. (2011). Some results for vacation systems with sojourn time limits. J. Appl. Prob. 48, 679-687. · Zbl 1226.60125 [7] Perry, D. and Asmussen, S. (1995). Rejection rules in the M/G/1 queue. Queueing Systems 19, 105-130. · Zbl 0835.60080 [8] Roubine, É. (ed.) (1970). Mathematics Applied to Physics . Springer, New York. · Zbl 0193.57201 [9] Takács, L. (1967). The distribution of the content of finite dams. J. Appl. Prob. 4, 151-161. \endharvreferences · Zbl 0164.50202
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.