Article: Analysis of MAP/PH1, PH2/1 retrial inventory queueing system with phase type vacation, working breakdown, repair, essential and optional inventories Journal: International Journal of Mathematics in Operational Research (IJMOR) 2026 Vol.33 No.3 pp.287 - 326 Abstract: This research analyses an inventory retrial queueing system with a constant retrial rate, working breakdown, repair, single vacation and inventory items that are essential and optional. We assume that an arrival follows the Markovian arrival process and that the server will provide them with phase-type services. The (s1, S1) and (s2, S2) policies are used to replenish the essential and optional inventory items, respectively, and there will be an exponential distribution in the replenishment time. The server takes a single vacation if there are no orbital customers, no essential inventory, or both. The server provides an optional item, if there exists a positive inventory level, otherwise the customer gets only the essential item and then leaves the system. When a server breaks down, the server provides the slow mode service for the current customer and then starts the repair process. The number of consumers in orbit, inventory level, and server status may all be determined in the steady state. Numerous key performance indicators are defined, and a cost analysis is obtained. In order to make our mathematical concept clearer, a few numerical examples are provided. Inderscience Publishers - linking academia, business and industry through research

Title: Analysis of MAP/PH1, PH2/1 retrial inventory queueing system with phase type vacation, working breakdown, repair, essential and optional inventories

Authors: S. Meena; G. Ayyappan

Addresses: Department of Mathematics, Puducherry Technological University, Puducherry, India ' Department of Mathematics, Puducherry Technological University, Puducherry, India

Abstract: This research analyses an inventory retrial queueing system with a constant retrial rate, working breakdown, repair, single vacation and inventory items that are essential and optional. We assume that an arrival follows the Markovian arrival process and that the server will provide them with phase-type services. The (s₁, S₁) and (s₂, S₂) policies are used to replenish the essential and optional inventory items, respectively, and there will be an exponential distribution in the replenishment time. The server takes a single vacation if there are no orbital customers, no essential inventory, or both. The server provides an optional item, if there exists a positive inventory level, otherwise the customer gets only the essential item and then leaves the system. When a server breaks down, the server provides the slow mode service for the current customer and then starts the repair process. The number of consumers in orbit, inventory level, and server status may all be determined in the steady state. Numerous key performance indicators are defined, and a cost analysis is obtained. In order to make our mathematical concept clearer, a few numerical examples are provided.

Keywords: queueing-inventory; (s, S) policy; retrial; essential and optional item; single vacation; working breakdown; repair; Markovian arrival process; MAP; phase-type distribution; matrix analytic method.

DOI: 10.1504/IJMOR.2026.152307

International Journal of Mathematics in Operational Research, 2026 Vol.33 No.3, pp.287 - 326

Received: 18 Feb 2024
Accepted: 21 Apr 2024
Published online: 16 Mar 2026 *

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Title: Analysis of MAP/PH1, PH2/1 retrial inventory queueing system with phase type vacation, working breakdown, repair, essential and optional inventories

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