A finite source retrial queueing inventory system with stock dependent arrival and heterogeneous servers

Abstract

This article discusses a finite-source stock-dependent stochastic inventory system with multiple servers and a retrial facility. The system can store a maximum of S items, and the lifetime of each item is exponentially distributed. The primary customer arrives at the waiting hall from the finite source and receives service from multi-servers. The rate at which customers arrive depends on the current stock level. If the waiting hall is full during the primary customer's arrival, he enters the finite orbit. Additionally, customers in the waiting hall may lose patience and enter the orbit. To replenish the stock, we follow the (s, Q) ordering policy. We calculate the joint probability distribution of the number of inventory items, busy servers, and number of customers in the waiting hall and orbit at a steady state. We conduct a comparative numerical analysis to determine the impact of heterogeneous and homogeneous service rates on various metrics, such as the average impatient customer rate, the fraction of successful retrials, and the average number of customers in the waiting hall and orbit.

Keywords: (s, Q) ordering policy; Classical retrial policy; Impatient customers; Markov chain; Multi-server; Steady-state; Stock-dependent arrival.

Fig. 1

Flow chart of the model.

Fig. 2

ETC for heterogeneous service rate: vs .

Fig. 3

ETC for heterogeneous service rate: S vs s.

Fig. 4

ETC for homogeneous service rate: vs .

Fig. 5

ETC for homogeneous service rate: S vs s.

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ETC for heterogeneous service rate: vs .

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ETC for heterogeneous service rate: vs .

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ETC for homogeneous service rate: vs .

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ETC for homogeneous service rate: vs .

Fig. 10

ETC for heterogeneous service rate: vs .

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ETC for heterogeneous service rate: c vs L.

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ETC for homogeneous service rate: vs .

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ETC for homogeneous service rate: c vs L.

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Average number of heterogeneous busy servers: vs .

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Average number of homogeneous busy servers: vs .

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Average number of heterogeneous busy servers: S vs s.

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Average number of heterogeneous busy servers: vs .

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Average number of homogeneous busy servers: S vs s.

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Average number of homogeneous busy servers: vs .

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Average number of heterogeneous busy servers: vs .

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Average number of heterogeneous busy servers: vs .

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Average number of homogeneous busy servers: vs .

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Average number of homogeneous busy servers: vs .

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Average number of heterogeneous busy servers: c vs L.

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Average number of homogeneous busy servers: c vs L.

Figure 26

Average number of customers in the waiting hall for heterogeneous service rate: vs .

Fig. 27

Average number of customers in the waiting hall for homogeneous service rate: vs .

Fig. 28

Average number of customers in the waiting hall for heterogeneous service rate: c vs .

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Average number of customers in the waiting hall for homogeneous service rate: c vs .

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Average number of customers in the waiting hall for heterogeneous service rate: vs .

Fig. 31

Average number of customers in the waiting hall for homogeneous service rate: vs .

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Average number of customers in the orbit for heterogeneous service rate: vs .

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Average number of customers in the orbit for heterogeneous service rate: c vs .

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Average number of customers in the orbit for homogeneous service rate: vs .

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Average number of customers in the orbit for homogeneous service rate: c vs .

Fig. 36

Average number of customers in the orbit for heterogeneous service rate: vs .

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Average number of customers in the orbit for homogeneous service rate: vs .

Fig. 38

Fraction of successful rate of retrial for heterogeneous service rate: vs .

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Fraction of successful rate of retrial for homogeneous service rate: vs .

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Fraction of successful rate of retrial for heterogeneous service rate: vs .

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Fraction of successful rate of retrial for homogeneous service rate: vs .

Fig. 42

Fraction of successful rate of retrial for heterogeneous service rate: c vs .

Fig. 43

Fraction of successful rate of retrial for homogeneous service rate: c vs .