Abstract
We study a G/GI/1 single-server queuing model with i.i.d. service times that are independent of a stationary process of inter-arrival times. We show that the distribution of the waiting time converges to a stationary law as time tends to infinity provided that inter-arrival times satisfy a Gärtner-Ellis type condition. A convergence rate is given and a law of large numbers established. These results provide tools for the statistical analysis of such systems, transcending the standard case with independent inter-arrival times.
Highlights
At the beginning of the 20th century, Agner Krarup Erlang, a Danish engineer did the first steps in a new branch of operations research what we call queuing theory [9]
Nowadays, queuing theory arises in many fields of engineering sciences such as inventory management, logistics, transportation, industrial engineering, optimal service design, telecommunication, etc
Among the most recent and interesting applications, we can mention a new queuing theory approach for cost reduction in product-service design [23]. Such solutions can be helpful for the service designer to find the optimized solution for the designed service
Summary
At the beginning of the 20th century, Agner Krarup Erlang, a Danish engineer did the first steps in a new branch of operations research what we call queuing theory [9]. If the inter-arrival times are i.i.d and the service times merely stationary and these two sequence are independent the process W is a Markov chain with driving noise Z in the random environment provided by S. The use of Gartner-Ellis-type conditions is not new in queuing theory: see e.g. Section 3 in [13] and [12], where the exponential tail of the limit distribution of the queue length is studied when the arrivals are weakly dependent. We recall a result on the ergodic behaviour of queuing systems with dependent service times which was obtained in Theorem 4.7 of [20]. As it is not easy to provide a clear-cut set of conditions for suitable Z, we refrain from these ramifications here
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