Queues With Impatient Customers And System Catastrophes

In the classic queuing model,customers arrive at the system and will definitely receive service before leaving,but in practical applications,impatient customers who leave before receiving service are very common.On the one hand,due to the excessive number of people in the system,customers may balk immediately before entering the system.On the other hand,customers may renege when they are not receiving services due to the long waiting time during the queue.In practical applications,catastrophes may also occur in the queuing system.When the catastrophes occur,the entire system fails and customers are emptied.The system began to repair,and normal operation started after the repair was completed.The queuing system with impatient customers and system catastrophes is an important direction of queuing theory research and is more in line with practical applications.Based on the study of classic M/M models,an M/M/1 queue with balking,reneging,catastrophes,server failures and repairs is considered.The arrivals follow a Poisson process and the servers serve according to an exponential distribution.When the number of customers is greater than or equal to the positive integer k,the customers start to balk and renege.By establishing the Kolmogorov differential equations,using the generating function technique,combining the Laplace transform and properties of Bessel functions,we obtain the transient and steady-state probability and other steady-state performance measures.We also extend the above model to a more general M/M/c model,where impatient behaviors occur when the number of customers is greater than or equal to c.