ORIGINAL ARTICLE Year : 2020  Volume : 8  Issue : 1  Page : 5356 Application of queuing analysis for optimized utilization of laboratory staff: An observational study Alekh Verma^{1}, Shakti Kumar Yadav^{1}, Aastha Narula^{1}, Amul K Butti^{1}, Namrata Sarin^{1}, Ruchika Gupta^{2}, Sompal Singh^{1}, ^{1} Department of Pathology, North Delhi Municipal Corporation Medical College and Hindu Rao Hospital, Delhi, India ^{2} Cytopathology Division, ICMRNational Institute of Cancer Prevention and Research, Noida, Uttar Pradesh, India Correspondence Address: Background and Aim: Queuing theory, a discipline of operation management, has seen great utility in various service industries. Application of queuing analysis (QA) in healthcare has been largely limited to emergency room, pharmacy, and patient appointment system. Utility of QA in the hospital laboratory has not been evaluated in detail so far. This study aimed at evaluating the application of QA at a report dispatch counter in the pathology department of our tertiarylevel hospital. Materials and Methods: In a crosssectional observational study, patient arrival at the report dispatch counter in the department of pathology was noted for 5 consecutive days while service rate at the same counter was observed for 4 days. QA was performed using Poisson distribution function for patient arrival and exponential distribution function for service rate. The expected waiting time in queue as well as number of patients waiting in the queue was calculated. Results: The average arrival rate (λ) was 6.94 and service rate (μ) was 8.34 patients, both for 5min interval periods. QA yielded the average waiting time in queue as 3.05 min. The expected number of patients in queue was estimated to be 4.26, implying that, on an average, four patients would be waiting in the queue to receive their report apart from the one being served at the a given time. Conclusion: QA can be efficiently applied to various areas of the hospital laboratory including report dispatch point. This is an extremely helpful tool to assist in staffing policy and assessing patient satisfaction at any patient contact point in the laboratory.
Introduction Queue is an inevitable phenomenon in our daily lives, resulting when demand exceeds serving capacity, at least at some point in time. A typical “Queuing system” comprises a population (finite or infinite), input stream of customers, queue, servers (one or more), and a queue discipline. The rate of arrival of customers in a system is denoted by λ and the rate at which they are served as μ. Queuing discipline can either be firstin firstout (FIFO) or firstinlastout, priority scheduling, or random order. Queuing theory (QT) is a scientific research methodology applying relevant mathematical modeling techniques to calculate various queue characteristics, such as average queue length, waiting time, and loss of customers or business.[1] Since its first application in the context of telephone facility about a century ago, QT has been successfully used in industrial settings for various operations. Although QT is an appropriate tool for healthcare settings as well, its application in a complex system such as a hospital has been limited to a few examples.[2] A recent report of application of queuing model to the emergency units and intensive care units (ICUs) of a general hospital modeled that increasing the number of physicians from one to two reduced the patients' waiting time in queue as well as the length of stay of a particular patient in either of the areas.[3] Although queuing analysis (QA) has been applied to hospital areas such as emergency room, pharmacy, outpatient department waiting area, and patient appointment system, the utility of this research technique in hospital laboratories has not been explored fully. Laboratory now form a vital point in the healthcare system with majority, if not all, patients undergoing some investigations. Hence, operations management through queuing analysis at laboratory point with patient contact is important to reduce patient waiting times. QA has been attempted in a laboratory's sample collection room for the assessment of staffing policy and evaluation of the scope of future expansion.[4] Khan and Callahan, in their modeling study, applied to a reallife laboratory situation demonstrated that QA could be utilized to determine the number of phlebotomists required for effective utilization of phlebotomy services of a laboratory with variable number of arriving patients.[4] Report dispatch counter is also a patient contact point in a laboratory that regularly witnesses a queue. Extensive search of the existing literature did not yield any report of application of QA to this area in a hospital. The present observational study attempts to apply the common QT models for the assessment of queue characteristics of a patients' queue at a report collection counter in the department of pathology of a tertiary level healthcare center. Materials and Methods This was a crosssectional, observational study conducted at the report dispatch counter in the department of pathology of our hospital. QA was performed, for which the following assumptions were considered: Single waiting line served by a single server (technical staff manning the counter)Arrival of patients for report collection follows a Poisson probability distributionService time follows an exponential probability distributionQueue discipline followed is FIFONo baulking (patient not joining the queue or looking for ways to shunt the queue) or reneging (patient in queue deciding to leave the service queue completely without being served). Arrival analysis The arrival of patients was studies for 5 consecutive working days during the period between 9 AM and 1 PM. The 4h period was divided into equal intervals of 5 min each. The observed frequency of the patient arrival was tabulated. The expected frequencies were calculated using the formula for Poisson probability distribution: [INLINE:1] Chisquare test was applied on the observed and expected frequencies to assess the goodness of fit of the probability distribution. Serving time analysis The time taken to serve each patient (i.e., issuing report) was noted for 4 consecutive days, and the results were tabulated as observed cumulative frequency. The expected cumulative frequencies were calculated using the formula for cumulative distribution function of exponential probability distribution, given below: Exponential probability distribution is calculated asP(t) = μe −μt for t ≥ 0. Cumulative distribution function:P (service time < t) =1 − e −μt. The observed and expected frequencies were analyzed using Chisquare test for assessing the goodness of fit. The mean arrival rate and service time were calculated using the collected data. QA was then performed to estimate the expected number of patients in the department as well as those waiting in the queue. The expected waiting times for a patient in the department and in the queue were estimated using the formulae: Expected number of patients in the department, L = λW, where L is number of customers in system and W is waiting time in the system. Expected number of patients in the queue, Lq = λWq, where L is number of customers in system and Wq is waiting time in the queue. Expected waiting time in the department, W = Wq + 1/λ. Expected waiting time in the queue, Wq = λ/[μ (μ − λ)], for single server mode. Probability of zero patients in the department, Pr = ρ0 (1− ρ), where ρ = λ/μ. Probability of n patients in the department, Pr (n) = ρn Pr (0). Results Over 5 consecutive days, 3913 patients visited our department for report collection. The observed frequencies for number of patients in 5min interval are tabulated in [Table 1]. The highest observed and expected frequency of the patients was seen for arrival of 5, 6, and 7 patients in a 5min interval. The average arrival rate (λ) was calculated to be 6.94.{Table 1} Chisquare test for observed and expected frequencies was statistically insignificant (P = 0.11), indicating that Poisson probability distribution is a good fit for the arrival of patients in our system [Figure 1].{Figure 1} The cumulative frequencies, both observed and expected, of the serving time of 2320 patients are tabulated in [Table 2]. The mathematical expectation yielded a result of 35.99 s, i.e., one patient was served in 35.99 s. This gives the average service rate (μ) of 8.34 patients in a 5min interval (1/35.99 × 300 s). The difference between observed and expected cumulative frequencies of service time was statistically insignificant (P = 0.128), suggesting goodness of fit of the exponential probability distribution for this function [Figure 2].{Table 2}{Figure 2} Queue analysis Using the mean arrival rate (λ) of 6.94, average service rate (μ) as 8.34, and number of servers as one, the average waiting time in the department (W) was 3.65 min while the time in the queue (Wq) was calculated to be 3.05 min. The average expected number of patients in the department (L) came out to be 5.10, and the expected patients in the queue (Lq) were estimated to be 4.26. This implies that apart from the patient being served at the counter at a given time, there would be an expected average of four patients in waiting for their turn in the queue. The calculated probabilities of there being 0, 1, or 2 patients in the department at any time were 0.16, 0.14, and 0.11, respectively, as tabulated in [Table 3].{Table 3} Discussion Laboratories form an integral and vital part of any medical establishment with responsibility of performing all testing and providing accurate and timely reports. Although majority of the private and many government hospitals in our city have moved to online report dispatch system, a few government hospitals still continue to have a physical report dispatch to the concerned wards or units in the hospital. In the physical dispatch system, the problem of missing or misplaced report is an unresolved issue and this forces the patients or their relatives to approach the laboratories in an effort to procure their test results. For this purpose, our laboratory operates a report collection counter where a queue of patients can usually be witnessed, especially in the morning hours. Although waiting in queues or lines is a phenomenon associated with our routine life, this can cause distress and lack of satisfaction in a hospital setting, given that the patients are already in a vulnerable position due to their illhealth. Hence, hospital managements are usually looking for ways to reduce this problem of queue in various departments and patient contact points in the hospital. QT, a discipline in the arena of operation management, has been applied extensively in the service industries for the assessment and improvement of staff schedules, working environment, productivity, etc. The use of QT in healthcare has been recent, gaining momentum in the last three decades. QT has been evaluated for emergency room arrivals, pharmacy, and outpatient department.[3],[5],[6] For instance, a study of QA in the emergency and ICU of a hospital showed that optimization of the number of physicians in such critical areas leads to reduction in patients' waiting time and length of stay in that department.[3] Similar planning can assist hospital administrators in planning for efficient utilization of capacity and infrastructure.[7] A study to assess the application of QT to patient satisfaction in a clinic with multiple singlechannel model and priority service showed that although majority of the patients were satisfied, suggestions such as arrival of doctors on time and change of service discipline to FIFO needed to be considered for higher satisfaction rate among the patients.[5] A study of QT using a simulator software in the emergency department concluded that addition of senior resident in emergency, additional bed in the ICUs and cardiac care units, augmentation of electrocardiogram service, and adding staff to laboratory and specialist consultation resulted in reduction in length of stay as well as the occupancy rate of nursing service in the hospital.[7] The application of QT in a hospital laboratory has not been studied in detail as yet. A study of the queuing network performance indicators in a central laboratory sample collection room of a hospital revealed undesirable queue characteristics during busy periods. The author suggested that increasing staff at the stage of sample labeling could reduce the average waiting time for the patients.[8] Khan and Callahan analyzed the primary data of a hospital laboratory using queuing analysis, operating characteristics of the existing system, and the potential for market expansion (considering laboratory as a cost center). The authors showed that it was possible to derive an equilibrium point for its full effective capacity with no reneging, leading to maximum revenue. The authors concluded that such a queuing model could be effectively utilized to derive the optimum facility size and corresponding workload, with flexibility to include the future growth projections.[4] Since ours is a governmentrun hospital where patients do not incur cost for investigations or management, we did not undertake a cost or revenue analysis in the present study. Extensive literature search failed to yield any report of application of QA to a laboratory's report collection room or counter. Undue long waiting times in a queue to receive their reports can cause distress and dissatisfaction among patients or their relatives and, in rare occasions, lead to verbal altercation between them and the laboratory staff. QA can be a helpful tool to avoid such situations by providing guidance on the staffing policy for such a counter on the basis of arrival rate and serving time for individual patients at different times during the day. In the present study, the average waiting time in the queue was calculated to be 3 min, which is well within the limits of tolerance for waiting in a queue. The average number of patients in the queue was four, apart from the one being served at the counter. These data suggest that our current staffing policy with regard to the report collection counter was appropriate for patient satisfaction. Conduct of this analysis at periodic intervals is likely to help in fine tuning the staffing during busy and lean months in a hospital or laboratory. Conclusion We demonstrate, for the first time, the application of QA to a laboratory report collection point. In the present study, our staffing policy at this point was found to be adequate. However, regular review of the workforce requirement at such patient contact points in the laboratory would help in assessment of staffing requirement at different patient arrival rates and, in turn, ensure maximum patient satisfaction within the available resources. Financial support and sponsorship Nil. Conflicts of interest There are no conflicts of interest. References


