Discrete-Time MMBP/Geo/1 Queues With Disasters

Discrete-time queuing systems with negative customers have received great attention due to their applications to the communication systems, manufacturing systems and other related areas. Since the concept of negative customers has first introduced by Gelenbe, it has been applied to various queuing system models including the M/M/1 queue, the M/G/1 queue, and the GI/M/1 queue.When a negative customer arrives at a queuing system, it removes a customer in service. On the other hand, disaster is one kind of negative customers, if a disaster arrives at a queuing system, it removes all present customers no matter it is in service or in queue. Queuing systems with disasters can well describe the virus outbreak in the computer networks and an infection of virus in email center.Most literature considers the customer arrivals have geometric distributions and without memory, however, measurements indicate that traffic in communication networks exhibits correlations. We use MMBP (Markov Modulated Bernoulli Process) to model the correlated arrival customers. Service times follow geometric distributions and only one customer can be served at a slot. Inter-arrival times also follow geometric distribution.This paper establishes and resolves three different discrete-time queuing systems, control of queuing delay in a buffer with time-varying arrival rate and impatient probability, discrete-time MMBP/Geo/1 queues with disasters, discrete-time MMBP/Geo/1 queues with disasters and repairs. Based on the probability generating function technique, expressions of some performance measures of the system such as average buffer contents and cell delay are given. Finally we give some numeric examples of performance measures.