On a queueing-inventory system with impatient customers, advanced reservation, cancellation, overbooking and common life time

We consider a queueing-inventory problem with Markovian arrival process, phase type distributed service time. The life time of items is exponentially distributed; all items perish together which means they have a common life time. Reservation (purchase) and cancellation of purchased items is permitted as long as the common life time is not realized. In addition overbooking of items upto a certain maximum is also permitted. When system is fully overbooked waiting customers tend to leave the system. On realization of common life time, instantly S items are procured resulting in the commencement of next cycle. In the particular case of Poisson process and exponentially distributed service time we show that the system admits asymptotic product form solution. System state characteristics are computed which are then numerically illustrated for different sets of values for the input parameters.