A perspective analysis of obligatory vacation and retention of impatient purchaser on queueing-inventory with retrial policy

The manuscript exemplifies a retrial queueing-inventory system with a maximum inventory level of $$S (=a \times n)$$ S ( = a × n ) units, where a and n are finite positive integers. It consists of an impatient purchaser’s retention, an obligatory vacation, and an ordinary vacation. It comprises a single Poisson arrival who demands exactly a single unit from inventory. Inventories are filled in accordance with the (s, Q) ordering policy. The server must take an obligatory vacation after serving each ‘n’ number of items, where ‘n’ is fixed. And the server takes an ordinary vacation once the server finds zero inventory after returning from its obligatory vacation. The purchasers may enter the orbit of infinite size with a prefixed probability when the server is on any kind of vacation. The impatient purchasers in the orbit have the decision of abandoning the orbit with a probability of $$p_1$$ p 1 or retaining it with a complimentary probability of $$q_1$$ q 1 . Vacations, replenishment, retention of impatient purchasers, and inter-retry duration are distributed exponentially. The stationary state probability vector is derived using the matrix geometric method. In the steady-state case, the joint probability distribution of the number of purchasers in the orbit and the inventory level is evaluated. Numerical computations have been used to determine the convexity of the overall expected cost rate for various ‘n’ values. It is based on the results of the cost factors. Some effects of retention and the obligatory vacation in the system are scrutinized.