Introduction

The recent outbreak of the 2019 novel Coronavirus or Severe Acute Respiratory Syndrome Coronavirus 2 that originated from Wuhan City in China has assumed global epidemic proportions. In such situations, timely delivery of services in the existing limited infrastructure is paramount to ensure service is still rendered to all those who require it. ^{Reference Mawonike and Mahachi1} Although the mortality rate in this pandemic is low, the disease is highly infectious. ^{Reference Lake2} Due to the possibility of asymptomatic carriers, public health measures including avoiding close contact with individuals having respiratory symptoms, frequent hand washing and social distancing have to be implemented. ^{Reference Lai, Shih, Ko, Tang and Hsueh3} Since health services constitute an essential component of control measures against such infectious agents, these public health measures apply equally to the health care workers and individuals visiting the health centers. It also needs to be kept in mind that the infrastructure (space), facilities (testing, isolation, and treatment), and personnel (doctors, nurses, paramedics) in any given health facility are finite leading to queues and these cannot be increased overnight to combat such a sudden epidemic or pandemic. Hence, mathematical modeling techniques come in handy to assist the hospital management and policymakers in ensuring optimum and judicious use of the available infrastructure and manpower while complying with the disease containment measures. Queuing analysis is one such modelling technique that can be useful in an appropriate allocation of personnel (infrastructure being non-modifiable) to reduce the queue length and ensuring that no patient is refused service. ^{Reference Mawonike and Mahachi1} Queuing models include certain assumptions for the arrival rate and service rate of a system and allows mathematical derivation of the number and type of servers required. ^{Reference Hassan, Shalbaf, Bilal and Rashid5} A typical queuing system comprises of the population from where customers (patients) arrive, number of customers arriving per unit of time, queue length restriction if any, number of servers (physicians), and the queue discipline used. The queue discipline is the order in which the customers or patients in a queue are attended to. The discipline is usually first-in-first-out; though triage is also used in emergency rooms of hospitals. ^{Reference Mawonike and Mahachi1} The queuing model to be applied in a given situation depends on the assumptions, finite or infinite source of patients and the capacity (finite or infinite) of the system. A queuing model written as the ‘M/M/c/K’ includes the first M for arrival distribution (for example, the Poisson distribution describing the number of events in an interval of time), ^{Reference Hassan, Shalbaf, Bilal and Rashid5} the second M for service time (for example, the exponential distribution), c for number of servers and K for queue length restriction. Information about the population being finite or infinite and queuing discipline used may be appended, if required, to this nomenclature. The mathematical formulae used in queuing analysis are based on Little’s law, L = λW, where L denotes the average number of customers in the system, λ (lambda) stands for average arrival rate into the system and W is the average amount of time spent by a customer in the system.

Queuing models have been applied in a variety of fields such as service centers, airport terminals, manufacturing systems, and communication networks. In healthcare too, queuing analysis has been resorted to for understanding and mitigation of congestion at crucial areas such as emergency departments or intensive care units (ICU). ^{Reference Olorunsola, Adeleke and Ogunlade4} Hospitals usually present as an ‘infinite source model’ where the number of customers or patients are independent and their arrival is assumed to follow a Poisson process. In a Poisson process model for a series of events, the average time between each event is known but the exact timing is random, hence, the arrival of an event is independent of the event prior to it. ^{Reference Mawonike and Mahachi1} At the same time, most of the service points in hospitals work with a ‘finite capacity’ leading to ‘reneging,’ that is patients leaving the queue and hospital without being attended to and ‘balking’, where the patient registers displeasure of the waiting time and leaves the hospital. However, healthcare services, especially the ones at crucial points need to strive to keep the reneging and balking at a minimum, and ensure that majority and if possible all patients coming to their facility are attended to with minimal waiting. Hence, application of queuing theory and techniques can serve to improve the healthcare service delivery in the existing infrastructure by optimizing the manpower and facility utilization. In order to maintain the recommended distance between 2 patients in keeping with the WHO guidelines on physical distancing, the maximum capacity of the screening room in our hospital was fixed, hence in our queuing problem, the system has a limited capacity (K). As a result of this, the present study was aimed to identify the optimal number of physicians in the screening area, given the space constraint, to ensure that service is rendered to each patient entering into the system.