An M/G/1 Queueing Model with Bernoulli Catastrophes and Repair Times

We study a single-server queueing model of the $M/G/1$ type with so-called Bernoulli catastrophes. Customers arrive according to a Poisson process with rate λ and are queued in case they find the server busy or under repair. The queue is served in FIFO order. Each customer requires a Coxian-2 distributed service time with parameters (b,μ₁,μ₂). To this M/C₂/1 queueing model a Poisson stream of catastrophes is added with rate γ. Upon occurrence of a catastrophe with probability α the queueing system (the queued customers plus the customer in service) is emptied completely, and with probability 1-α a catastrophe only eliminates the customer in service (a so-called negative arrival). We consider two versions: (i) without repair times, i.e. occurrence of a catastrophe the server can immediately start a new service when a customer is or will be present, and (ii) with repair times, i.e. occurrence of a catastrophe the server starts an exponential repair time with rate ν. For both versions we study the steady-state behaviour by deriving the (partial) probability generating functions (PGFs) of the limiting probabilities of the system-states. From these PGFs explicit expressions for the probabilities of the system states will be deduced, as well as several other characteristics of the system, such as the long-run fraction of time that the server is idle, busy or under repair, the steady-state queue-length distribution, the waiting-time distribution of a so-called 'successful customer' (i.e. customer who will reach the server to start his service), the long-run fraction of these successful customers, and many more. Numerical results will give us further insight into the behaviour of this $M/G/1$ queueing model with Bernoulli catastrophes and repair times.