TheGrXn/Gn/∞ system: System size

An important property of most infinite server systems is that customers are independent of each other once they enter the system. Though this non-interacting property (NIP) has been instrumental in facilitating excellent results for infinite server systems in the past, the utility of this property has not been fully exploited or even fully recognized. This paper exploits theNIP by investigating a general infinite server system with batch arrivals following a Markov renewal input process. The batch sizes and service times depend on the customer types which are regulated by the Markov renewal process. By conditional approaches, analytical results are obtained for the generating functions and binomial moments of both the continuous time system size and pre-arrival system size. These results extend the previous results on infinite server queues significantly.

[1]  J. Templeton,et al.  On the GIX/G/∞ system , 1990 .

[2]  D. N. Shanbhag,et al.  On infinite server queues with batch arrivals , 1966 .

[3]  Gordon F. Newell,et al.  The M/M/⁰⁰ service system with ranked servers in heavy traffic , 1984 .

[4]  M. Neuts,et al.  The infinite server queue with semi-Markovian arrivals and negative exponential services , 1972, Journal of Applied Probability.

[5]  Edward G. Coffman,et al.  A Stochastic Model of Fragmentation in Dynamic Storage Allocation , 1985, SIAM J. Comput..

[6]  M. L. Chaudhry,et al.  On the service system MX/G/∞ , 1983 .

[7]  M. Kendall,et al.  The Advanced Theory of Statistics: Volume 1, Distribution Theory , 1978 .

[8]  L. Takács.  Queues with infinitely many servers , 1980 .

[9]  J. Templeton,et al.  On the GI X /G/ ∞ system , 1990 .

[10]  Erhan Çinlar,et al.  Introduction to stochastic processes , 1974 .

[11]  M. L. Chaudhry,et al.  A first course in bulk queues , 1983 .

[12]  Antoine-S Bailly,et al.  Science régionale - Walter Isard, Introduction to régional science. Englewood Cliffs (NJ), Prentice-Hall, 1975 , 1975 .

[13]  E. Çinlar.  Markov renewal theory , 1969, Advances in Applied Probability.

[14]  Fumiaki Machihara.  AN INFINITELY-MANY SERVER QUEUE HAVING MARKOV RENEWAL ARRIVALS AND HYPEREXPONENTIAL SERVICE TIMES , 1986 .

[15]  H. P. Galliher,et al.  Dynamics of Two Classes of Continuous-Review Inventory Systems , 1959 .

[16]  L. Takács,et al.  On a coincidence problem concerning telephone traffic , 1958 .

[17]  P. Purdue,et al.  An infinite-server queue subject to an extraneous phase process and related models , 1981 .

[18]  Woollcott Smith.  Technical Note - The Infinitely-Many-Server Queue with Semi-Markovian Arrivals and Customer-Dependent Exponential Service Times , 1972, Oper. Res..