| Summary: | This article analyses a four-dimensional stochastic queueing-inventory system with multiple server vacations and a state-dependent arrival process. The server can start multiple vacations at a random time only when there is no customer in the waiting hall and the inventory level is zero. The arrival flow of customers in the system is state-dependent. Whenever the arriving customer finds that the waiting hall is full, they enter into the infinite orbit and they retry to enter the waiting hall. If there is at least one space in the waiting hall, the orbital customer enters the waiting hall. When the server is on vacation, the primary (retrial) customer enters the system with a rate of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>λ</mi><mn>1</mn></msub><mrow><mo>(</mo><msub><mi>θ</mi><mn>1</mn></msub><mo>)</mo></mrow></mrow></semantics></math></inline-formula>. If the server is not on vacation, the primary (retrial) arrival occurs with a rate of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>λ</mi><mn>2</mn></msub><mrow><mo>(</mo><msub><mi>θ</mi><mn>2</mn></msub><mo>)</mo></mrow></mrow></semantics></math></inline-formula>. Each arrival rate follows an independent Poisson distribution. The service is provided to customers one by one in a positive time with the rate of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>μ</mi></semantics></math></inline-formula>, which follows exponential distribution. When the inventory level drops to a fixed <i>s</i>, reorder of <i>Q</i> items is triggered immediately under <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>s</mi><mo>,</mo><mi>Q</mi><mo>)</mo></mrow></semantics></math></inline-formula> ordering policy. The stability of the system has been analysed, and using the Neuts matrix geometric approach, the stationary probability vectors have been obtained. Moreover, various system performance measures are derived. The expected total cost analysis explores and verifies the characteristics of the assumed parameters of this model. The average waiting time of a customer in the waiting hall and orbit are investigated using all the parameters. The monotonicity of the parameters is verified with its characteristics by the numerical simulation. The discussion about the fraction time server being on vacation suggests that as the server’s vacation duration reduces, its fraction time also reduces. The mean number of customers in the waiting hall and orbit is reduced whenever the average service time per customer and average replenishment time are reduced.
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