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Probability, Statistics and Queueing Theory

As an example of Kendall’s notation, the expression M/G/1 – LCFS preemptive resume (PR) describes an elementary queueing system with exponentially distributed interarrival times, arbitrarily distributed service times, and a single server. The queueing discipline is LCFS where a newly arriving job interrupts the job currently being processed and replaces it in the server. The servicing of the job that was interrupted is resumed only after all jobs that arrived after it have completed service. M/G/1/K/N describes a finite-source queueing system with exponentially distributed source times, arbitrarily distributed service times, and a single server. There are N request in the system and they are accepted for service iff the number of requests staying at the server is less than K. The rejected customers return to the source and start a new source time with the same distribution. It should be noted that as a special case of this situation the M/G/1/N/N system could be considered. However, in this case we use the traditional M/G/1//N notation, that is the missing letter, as usual in this framework, means infinite capacity, and FCFS service rule. It is natural to extend this notation to heterogeneous requests, too. The case when we have different customers is denoted by →. So, the M / ~ G/ ~ 1/K/N denotes the above system with different arrival rates and service times.