Transient analysis of a queue with system disasters and customer impatience

Abstract

A single server queue with Poisson arrivals and exponential service times is studied. The system suffers disastrous breakdowns at an exponential rate, resulting in the loss of all running and waiting customers. When the system is down, it undergoes a repair mechanism where the repair time follows an exponential distribution. During the repair time any new arrival is allowed to join the system, but the customers become impatient when the server is not available for a long time. In essence, each customer, upon arrival, activates an individual timer, which again follows an exponential distribution with parameter ξ . If the system is not repaired before the customer's timer expires, the customer abandons the queue and never returns. The time-dependent system size probabilities are presented using generating functions and continued fractions.

A transient solution to the M/M/c queuing model equation with balking and catastrophes

“…He derived a transient solution to the model. Sudesh [39] studied the transient behavior of a single server queue with disasters and impatient customers. Kumar and Sharma [27] considered the negative impact of customer impatience and incorporated the probability of retaining a reneging customer into an M/M/1/N queuing system with reneging.…”

Analysis of the M/G/1 queue in multi-phase random environment with disasters

“…In succession, Chakravarthy [7] generalized the model to Markovian arrivals and obtained the steady state probabilities by the matrix analytic approach. Sudhesh [23] gave an explicit transient solution for the state probabilities of the same model studied in [27]. Baumann and Sandmann [2] studied a model about level dependent quasi-birth-and-death processes with catastrophes, and gave a matrix analytic algorithm to analyze M/M/c queues in a random environment with catastrophes and state dependent rates.…”

On Two Modifications of Er/Es/1/m Queuing System Subject to Disasters

“…Whenever a catastrophe occurs the system empties instantly and all newly arriving customers are lost during the server repair. Sudhesh [2] studied a similar M/M/1 queue which differs in the fact that customers entering the system become impatient when the server is down. The authors of papers [1] and [2] executed transient analysis of the systems, which means they derived formulas for system state probabilities as functions of the time t. Yechiali [3] examined an M/M/c queue with random disastrous failures which cause all present customers to be lost.…”