The restricted Erlang-R Queue: Finite-size effects in service systems with returning customers

TL;DR

This paper analyzes a restricted Erlang-R queue model with capacity constraints, providing approximations and resource allocation strategies for healthcare systems with returning customers and space limitations.

Contribution

It introduces a finite-size Erlang-R model with blocking and holding policies, offering explicit capacity allocation methods and insights into system dynamics with returning customers.

Findings

Developed many-server approximations for performance measures

Derived explicit capacity allocation methodology

Validated policies through hospital case studies

Abstract

Motivated by health care systems with repeated services that have both personnel (nurse and physician) and space (beds) constraints, we study a restricted version of the Erlang-R model. The space restriction policies we consider are blocking or holding in a pre-entrant queue. We develop many-server approximations for the system performance measures when either policy applies, and explore the connection between them. We show that capacity allocation of both resources should be determined simultaneously, and derive the methodology to determine it explicitly. We show that the system dynamics is captured by the fraction of needy time in the network, and that returning customers should be accounted for both in steady-state and time-varying conditions. We demonstrate the application of our policies in two case studies of resource allocation in hospitals.