Quasi - Birth - Death Processes

1 Description of the Model This case study is analog to the one described in [2]. We considered a system consisting of a fixed number of m processors and an infinite queue for storing job requests. The processing speed of a processor is described by the rate γ, while λ describes the incoming rate of new jobs. If a new job arrives while at least one processor is idle, the job will be processed directly. Otherwise, it will be put into a waiting queue. If there are idle processors and the waiting queue is non-empty, a job will be taken from the queue and processed immediately. To model this spontaneous transition, a rate µ λ is used. The stochastic Petri net (SPN) used in [2] is depicted in Figure 1 for m = 3. Tokens in Figure 1: Stochastic Petri Net of the model for m = 3 place p 1 represent the number of idle processors, place p 2 describes the number of busy processors and place p 3 gives the number of jobs in the queue. Transition t 1 models the case of an incoming job given that at least one processor is idle, whereas t 4 describes the case in which all processors are busy, thus the job is put into the queue. Transition t 2 represents the successful termination of a job. Finally, t 3 is the spontaneous transition in 1