A matrix-analytic queueing-inventory model of community pharmacy service: server interruption, lost sales, and optimal safety stock in Nigerian patent medicine stores

Abstract

In Nigerian patent medicine stores and community pharmacies, a single pharmacist often doubles as dispenser and stock-keeper: when a drug's availability is uncertain, the pharmacist must step away to check the back store, halting the queue; if the drug is unavailable, the customer leaves without buying (a lost sale), while others may abandon the queue (reneging) or never join it (balking). This paper models the system as a continuous-time Markov chain coupling a finite-capacity M/M/1/N queue with state-dependent balking and reneging to a continuous-review (r,Q) inventory policy with lost sales for a high-turnover drug. Service is represented as a two-phase process (direct dispensing, or a backroom check followed by dispensing) yielding a closed-form effective service rate as a function of the probability a check is required, a proxy for stock-record and layout quality. The joint queue-inventory-service process is formulated as a quasi-birth-death chain and solved via matrix-analytic methods. A numerical study shows that raising this probability from 0.05 to 0.30 cuts the effective service rate by 37.5%, increases mean time in the pharmacy from 14.3 to 21.2 minutes, and more than doubles reneging. Counter-intuitively, the cost-minimizing safety stock falls under poor organization, since congestion, not stockouts, becomes the binding constraint, inventory and service-organization decisions must be optimized jointly, not separately.