Queueing Models and Service Management

A Batch Demands Queueing-inventory Model with Positive Replenishment Time

Akash Verma — 2025-03-01

This study examines an M/M/1 queueing model using a (s,Q) inventory system and batch demand environment. The customers may request one or more items with a maximum demand of B items. There is just one service provider and a limited number of places available for customers to wait. A customer who arrived in accordance with the Poisson process is forced to leave without service if the waiting space is full. The service time and the replenishment time are assumed to have independent exponential distributions. It is assumed that maximum batch demand B is smaller than or equal to reorder level s in order to avoid repeated replenishment in a service. With the help of an iterative process, we are able to derive the steady-state joint probability distribution of the number of customers in the system and the on-hand inventory level of the queueing-inventory model. The optimum values for waiting space (N), reorder level (s), and order quantity (Q) are found by establishing a number of stationary system performance measures and estimating the total expected cost function under an appropriate cost structure. Some numerical results for various model parameters are provided in order to explain the key performance measures of the system. We execute simulation results with ARENA software to validate our model. We also perform simulation results for the equivalent M/G/1 queueing-inventory model.

Analysis of a Dual Tandem Queue with Servers Floating Between the Stages

Sergei Dudin — 2025-03-01

A tandem queueing system with a correlated arrival process and two multi-server stages is analyzed. The capacity of the buffer at stage 1 is infinite. The capacity of the bufferat stage 2 is finite. The total number of available servers is fixed. Servers are dynamically shared between the stages in such a way that any server cannot stay idle if at least one of the buffers is not idle. Servers can transit between the stages only at service completion epochs. Servers from stage 2 can transit to stage 1 even if the buffer at stage 2 is not empty, according to the control policy defined by two integer thresholds. If service by the server assigned to stage 2 is completed when the number of customers in the stage 1 buffer is not less than the first threshold and the number of customers in the stage 2 buffer is less than the second threshold, the released server is re-assigned to stage 1 and immediately starts service. If service by a server assigned to stage 1 is completed when the stage 2 buffer is full, the released server is re-assigned to stage 2 and immediately starts service. In the case of service completion at stage 2 during the epoch when the buffers at both stages are idle, the released server is re-assigned to stage 1 and waits for a new customer arrival at this stage. Customers’ arrival is described by the Markov arrival process (MAP). Each customer has to receive service at both stages of the tandem or only at stage 1. The service times at both stages have a phase-type distribution with parameters depending on the stage. Under the fixed values of the thresholds, analysis of the stationary behavior of the tandem is implemented, including derivation of the ergodicity condition, computation of the stationary distribution of the number of customers at each stage, and derivation of expressions for the key performance indicators. Analysis is essentially based on the proper use of the notion of the generalized phase-type distribution. The results of numerical experiments illustrating the feasibility of the proposed algorithms and highlighting the dependence of the performance measures of the system on the parameters of the control policy are presented. The problem of the optimal choice of thresholds is briefly considered.

Applications of Queueing Models to Improve Airport Operations

Shuzhen Sun — 2025-03-01

Balancing the waiting times of passengers and the allocation of personnel and facilities among different operations in an airport is a challenging problem. Queueing models have been widely regarded as useful tools for evaluating the performance of a service system because of their ability to provide quick and reasonably accurate values for performance measures related to waiting times and queue lengths. Hence, the queueing approach is an important tool to analyze and improve airport operations. This paper reviews and categorizes the literature on the application of queueing theory to model the check-in, security screening, and baggage claim processes at passenger terminals and the runway service, taxi-out, and landing processes at aircraft terminals. In doing so, the paper also identifies potential future research opportunities, some of which are motivated by new airport technologies and processes.