1. INTRODUCTION:

The theory of queues was first initiated by the Danish mathematician A.K. Erlang [1] some important concepts like single and multiple queues in Queuing theory. Later on Arnoud, M. deBruin, A.C. Van Rossum and et. al [2] studied that the strong focus on raising occupancy rates of hospital management is unrealistic and counterproductive. Economies of scale cannot be neglected. An important result is that refused admissions at the First Cardiac Aid (FCA) are primarily caused by unavailability of beds downstream the care chain. Both variability in length of stay (LOS) and fluctuations in arrivals result in large work load variations. After that R. A. Adeleke, O. D. Ogunwale and et. al [3] formulated and adopted to make the government established hospitals, primary health care centers, federal medical centers, university teaching hospitals and so on functional effectively. These health policies also brought about the establishment of global and local health organization such as World Health Organization (WHO).

United Nations International Children Emergency Fund (UNICEF) and a host of others. And then Reetu Mehandiratta [4] attempted to analyze the theory (Queuing)and instances of use of queuing the oryin health care organizations around the world and benefits acquired from the same. Ekpenyong, Emmanuel John and Udoh, Nse Sunday [5] proposed some models which improves on the performance measures of the Single-Server Single Queue System with Multiple Phases. The extension results in a new Queuing System of Multi- Server with Multiple Phases under the conditions of First Come First Served, infinite population source, Poisson arrivals and Erlang service time. Queuing properties such as expected total service time, its variance and some performance measures like the expected number of phases in the system, expected number of phases in the queue, expected number of customers in the queue, expected waiting times in the queue and in the system as well as the number of customers in the system have been derived in this work for this Multi-Server with Multiple Phases (M/Ek/s: (∞/FCFS)) queuing model with k identified stages in series. These performance measures so obtained were compared with those of the already existing Single-Server with Multiple Phases (M/Ek/1:( /FCFS)model. Prasanta Kumar Brahma [6] introduced some models using single and multiple servers which reveals a preference for a multi-channel system determines limits to its usage and makes recommendations for improvement in service delivery. And then Avishai Mandelbaum, Petar Momcilovic [7] proposed the model G/GI/N+GI, is considered in the quality-and efficiency - driven (QED) regime. QED performance entails short waiting times and scarce abandonment’s (high quality) jointly with high servers’ utilization (high efficiency), which is feasible when many server scater to a single queue. For the G/GI/N+GI queue, we derive diffusion approximations for both its queue-length and virtual-waiting-time processes. Kembe, M. M., Onah, E. S, et. al [8] studied some concepts on Queuing theory. In their study, the queuing characteristics at the Reverside Specialist clinic of the Federal medical centre, Fatma Poni Mardiah, Mursyid Hasan Basri [9] were mainly concentrated on outpatient services. Outpatient services have become an important component of healthcare. By hide bound thinking, the medical profession emphasized that a physician’s time is more valuable than a patient’s time. Consequently, the appointment system was designed to minimize physician’s idle time overlooking patients waiting time. This is no longer valid in today’s consumer oriented society. Long waiting times for treatment in the outpatient department followed by short consultations has long been a complaint. Nowadays, customers use waiting time as a decisive factor in choosing a service provider. Therefore, idle time of both parties must be considered in designing an appointment system although these two objectives are contradicted to each other. They aims to provide a study of the major causes of patients length of time for medical treatment in an outpatient clinic a to neo f Indonesian public hospital and also provide recommendation on the best strategy to improve the appointment system so that can maximize the effectiveness and efficiency of resource and capacity. The hospital queue model use single-channel multi phase systems. Queuing theory be the first tool to look at patient waiting times on each server independently. The results show that the hospital should change the appointment system for physicians. Applying doctor on call system may appear to reduce doctor’s idle time but lead to high patients’ waiting times. In some cases, the appointment system make doctor to be back and for that other hospital, so it was not directly affect the productivity of a doctor. Not only construct the appointment system, they should take attention of patient flow and set scheduling of the capacity to increase the effective and efficiency outpatient department performance. Is an P Lade, Sandeep A Chowriwar et. al [10] worked on some models which helps to the scheduling system of the department. Queuing the or y can be used to predict some of the important parameters like total waiting time, average waiting time of patients, average queue length. Olorunsola S. A., Adeleke R.A., et. al [11] studied Queuing analysis of patient flow in hospitals. In their study they discussed that waiting on a queueis not usually interesting, but reduction in this waiting time usually requires planning and extra investments. The increasing population and health-need due to adverse environmental condition shave led to escalating waiting times and congestion in hospitals especially in the Emergency and Accident Departments (EAD).It is universally acknowledged that a hospital should treat its patients, especially those in need of critical care, in timely manner. Incidentally, this is not achieved in practice particularly in government owned health institutions because of high demand and limited resources in these hospitals. To enhance the level of admittance to care, optimal beds required in hospital is needed and this can be achieved by adequate knowledge of patient flow. Saima Mustafa, S. unNisa [12] focused on the applications of queuing theory in the field of health care (hospital) i.e. one of the biological paradigms. The health systems should have an ability to deliver safe, efficient and smooth services to the patients. We considered different departments of health care sector such as patient’s registration department, outpatients department and pharmacy department and also observed different processes in the system by queuing models. Different queues and numbers of servers involved in the processes are also observed by using appropriate probability distributions. The arrival process calculated by exponential distributions and service process is measured by Poisson distributions. Single server M/M/1 and multiple server queuing models M/M/2are used in order to analyze queuing parameters and performance measures of the system. Queuing analysis is done by using queuing simulation technique in order to compute the values of unknown parameters and performance measures. Besides this, strength of relationship between different performance measures is calculated.

2. Modeling the emergency cardiac care chain [2]

The model which has been developed for the primary goal of this study. First, in Section2.1 the phenomena of blocking and economies of scale are introduced. Both blocking (e.g. refused admissions) and economies of scale are important features of health care processes and are directly related to the quality of care. Ridge et al. [13] also describe the non-linear relationship between number of beds, mean occupancy level and the number of patients that have to be transferred through lack of bed space.

2.1 The phenomenon of blocking and impact of economies of scale on occupancy rates

An important model from queuing theory is the in Kendall’s notation. In this model customers (for example patients) arrive according to a Poisson process with intensity λ. This is the real demand, thus including the refused admissions. The LOS of arriving patients is independent and exponentially distributed with expectation μ. The number of beds is equal to C. There is no waiting area, which means that an arriving patient who finds all beds occupied is blocked. In real-life the consequence of blocking could well be a refused admission.

This is a more realistic representation of emergency in- patient flow. The fraction of patients which is blocked and sent away to other hospitals in the long run P_c can be calculated with the Erlang loss formula,

----------------------(1)

The occupancy rate is related to the real demand and LOS and can be defined as --- (2)

The term can be entitled as the effective demand as they refused admissions are subtracted from the real demand. Furthermore, the product is known as the work load of the system. Many hospitals use the same target occupancy rate for all hospital units, no matter the size of the unit. In general the unit size varies between 6 and approximately 60 beds. The target occupancy rate Is typically set at 85% and has developed into a golden standard [14]. The feasibility of this target is no matter of discussion in the considered hospital.

In order to demonstrate the relation between the size of a hospital unit, the feasibility of the 85% target and the fraction of refused admissions two calculations have been made. Both calculations have been performed via iteration of Eqs. 1 and 2.

1. The percentage of refused admissions (Pc) given an occupancy rate (ρ) of 85% (2≤c≤60)

2. The target occupancy rate (ρ) for Pc =0.05(5%refused admissions) (2≤c≤60).

The conclusion is clear and important. Larger hospital units can operate at higher occupancy rates than smaller ones while attaining the same percentage of refused admissions. Therefore, one target occupancy rate for all hospital units is not realistic. The 85% target is only attainable for units with more than 50beds, assuming Pc =0.05 is acceptable. If we hold the 85% target for a small unit such as the CCU (6beds) nearly half of all arriving patients is blocked.

Currently, the discussion about refused admissions does not focus on the direct relation between the size of a hospital unit and the feasibility of target occupancy rates.

3. METHODOLOGY [3]

Data on arrival times, time service begins, time service ends, and departure time of 100 patients was collected over 14days. This data will enable us to obtain the arrival rate, the service rate, and the traffic intensity of the patients using results from the birth and death model (which is synonymous to arrival and departure).

3.1 Model Specification

Them/m/1 Queue (Single-Channel Queuing System). In this queuing system, the customers arrive according to a Poisson process with rate. The time it takes to serve every customer is an exponential random variable. The service times are mutually independent and further independent of the inter arrival times. When a customer enters an empty system, his service starts at once and if the system is non-empty, the incoming customer joins the queue when a service completion occurs, a customer from the queue if any enters the service facility at once to get served.

4. Health service capacity planning [4]

It is common for health care managers to project workload for physical infrastructure and manpower planning. This may be done at different departments, hospitals or even national level. It is common method to look at past trends, estimate the historical year-on-year growth and extrapolate this growth rate to the future. However there are two potential problems. Firstly we seldom see definitive trend and the estimation of ‘growth rate’ is highly dependent upon start and end points of the time intervals. Secondly, the assumption of long lasting trend is also unrealistic. A health care utilization is often closely related to age, a more robust way to project is to use population based drivers. We can first drive the age specific utilization rate, which is the number of encounters (e.g: emergency or patient attendance, hospital admissions) as per population specific to each age group.

4.1 Queuing theory and health care

The health systems should have an ability to deliver safe, efficient and smooth services the patients. Several key reimbursement changes, increasing critiques and cost pressures on the system and increasing demand of quality and efficacy from highly aware and educated patients due to advances in technology and telecommunications have started putting more pressure on the health care managers to respond to these concerns. Queuing theory is an example of the use in healthcare. It essentially deals with patient flow through the system, if patient flow is good then patient queuing is minimized, if it is bad then the system may suffer loss of business and patients may suffer considerable queuing delays. Health care system can be visualized as a complex queuing network in delays can be reduced through the following ways:

(a) Synchronization of work among service stages (e.g., coordination of tests treatments, discharge process)

(b) Scheduling of resources (e.g., doctors and nurses) to match patterns of arrival.

(c) Constant system monitoring (e.g., tracking number of patients waiting by location, diagnostic grouping and acuity) linked to immediate actions.

5. Model [5]

The model is made up of‘s’ multiple service channels with ‘k’ identical stages in series, each with average time of.The distribution of total servicing time of customers in the system is some joint distribution of time in all these stages. Customers arrive in a single queue and a set of them, whose number is based on the number of servers in each phase, enters the system to be served in the first phase before proceeding to the second phase, up to the k th phase. The assumptions are that each set of customers is served in k-phases set-by-set and a new service does not start until all k-phases have been completed. Moreover, the queue discipline is first come first served with infinite source. The arrivals follow a Poisson distribution and the service times follow Erlang distribution. Symbolically, the model can be expressed as

5.1 Determination of Expected Total Service Time and its Variance.

Let denote the number of customers served per unit time, and then will be the number of phases served per unit time. The probability density function for Erlang distribution is;

With variance .

6. CHARACTERISTICS OF MULTI-SERVER SYSTEM [6]

According to Safe Associates (2002), a multi-server system has all the features of simple queue and in addition assumes no limit to the permissible length of the queue. Also, all servers are assumed to perform at the same rate.

6.1 METHODOLGY OF THE STUDY

Indian Hospital Industry most be devilled with patients waiting problems Is studied here for a period of four weeks in IMS and SUM Hospital; through observation, interview and questionnaire administration. The variables measured include arrival rate () and service rate (). These are analyzed for simultaneous efficiency inpatient satisfaction and cost minimization through the use of a single channel and multi-channel queuing models, which are compared for a number of queue performances such as; the average time each patient spends in the queue and in the system, average number of patients in the queue and in the system and the probability of the system being idle. In the realization of these objectives, primary data in respect of patient’s arrival rate and service rate were used and were obtained through observations while patient attitude survey was carried out through a total of fifty questionnaires per week administered on two hundred patients randomly selected for such purpose. These activities were carried out for four weeks ( December 2009) that hosted four principal festivities; Muslim Sallah, Xmas, end of a year and the beginning of another year and thereby considered the most appropriate for this study.

7. The M/M/S Model [7]

Data for this study were collected from riverside specialist clinic of federal medical care centre Mukurdi. The methods employed during data collection were direct observation and personal interview and questionnaire administering by the researcher. Data were collected for 4 weeks. The following assumptions were made for queuing system at the Riverside specialist clinic which is in accordance with the queuing theory.

(a)Arrivals follow a Poisson probability distribution at an average rate of customers (patients) per unit of time.

(b) The queue discipline is First-Come, First-Served (FCFS) basis by any of the servers. There is no priority classification for any arrival.

(d) There is no limit to the number of the queue (infinite).

(e) The service providers are working at their full capacity.

(f) The average arrival rate greater than average service rate.

(g) Servers here represent only doctors but no other medical personnel’s.

(h) Service rate is independent of the line length; service providers do not go faster because the line is longer.

8. Outpatient management [8]

Outpatient services are gradually becoming an essential component in health care. The development of the times, technology and the rapid increase in population make the theories need to be developed. The objective of outpatient scheduling is to and an appointment system for which a particular measure of performance is optimized in a clinical environment.

8.1 Data calculation and analysis.

Descriptive analysis: The hospital has two kinds ambulatory service. One service is general outpatient clinic that was treated by residency. The others are Specialist outpatient clinic that was treated by specialist (eg. internist, ophthalmologist, and obstetrician).The studied was conducted in specialty outpatient clinic. This clinic has eleven specialists, there are; internal medicine, obstetrics and gynecology and venereology, neurology, medical rehabilitation, pediatric, psychiatry and ophthalmology. Observations made during a month. After the observation; it was decided for data retrieval done on Monday and Tuesday, which is on the busy days of the week. The data used are the number of patient arrivals per specialist per day, arrival time of doctor and patient, length of registration service and the duration of examination.

Outpatient waiting time:

The ambulatory facilities are designed to make more effective use of patients and doctors time the queuing formula for model single channel used was:

Average server utilization , where = Arrival rate and = service rate

Average number of the customers in the queue

Average number of the customers in the system

Average waiting time in the queue

Average waiting time in the system

Based on the calculations for the first server using one month daily data, the average time need on waiting for the patient is 0.3hours, or 18 minutes.

9. AVERAGE WAITING TIME FOR OPD SECTION WITH ONE EXTRA DOCTOR [9]

In order to reduce the average waiting time of OPD patients’ one extra doctor is made available and the waiting time calculations are done. The results obtained are consolidated. It can be identified that addition of one extra resource or say a Doctor to OPD section has reduced down the 4.25 min. This indicates that the addition of doctor has reduced down the waiting time to much lower level.

The peak waiting hours after allocation of 1 more doctor to the OPD section. The average waiting time has been reduced down to 4.25 min the highest waiting time at the peak is 14 min.

9.1 Results Discussions for the OPD Section

The calculations done for the OPD section with 4 doctors and 5 doctors indicate that the addition of one more doctor has reduced the average waiting time from 7.20 min to 4.25 min it can be concluded that the peak waiting time is 27 min when 4doctors are available and its 14 min for 5 doctors. Generally the waiting time is more in morning session, i.e., from 9 a.m. to 12 noon. From the calculations, it’s about 0 to 150 min when the waiting time is very high.

10. Length of stay distribution [10]

The number of days in hospital for a patient is described by the length of stay (LOS).

LOS is defined as the time of discharge minus time of admission. The average length

Of stay in hospitals is a statistical calculation often used for health planning purpose.

Average length of stay (in days) =Total inpatient days of care/Total admissions.

The following definitions for each of the four data items included in the above calculations:

Total discharge days: The sum of days spent in the hospital for each inpatient who was discharged during the time period examined regardless of when the patient was admitted.

Total discharges: The number of inpatients released from the hospital during the time period examined.

Total inpatient days of care: Sum of the each daily inpatient census for the time examined.

Total admissions: The number of individuals formally accepted into inpatient units of the hospital during the period examined.

11. MATERIALS ANDMETHODS [11]

Data are taken for two days of a week from a public hospital of Rawalpindi. Three departments of this hospital are focused which are “patient’s registration department”, “out- patients department” and “pharmacy department”. Data is collected only from patients and doctors and not any other medical personals in outpatient department. The data collection is primary. The method are used for data collection are, “questionnaire”, “direct observation” and “interviews”. The following assumptions are fulfilled by the data for queuing models.

1. It is assumed that the patients’ on set follow a Poisson probability distribution.

2. Time of inter-arrival of patients is in dependent and exponentially distributed.

3. Service time is also exponentially distributed.

4. It is assumed that patients are served by any server on first-come first-served basis.

5. It is also assumed that no patient will leave the queue without getting service.

6. The queue is infinite.

7. For outpatient department doctors were only servers.

8. Rate of serving was not dependent on the queue length. Serving rate remained moderate despite of queue length.

11.1 Patient’s Registration Department and Queuing Model

In any hospital before being paid services from doctors patient must have to register himself from front office i.e. patients registration window. This window is provided a computer slip to the patient of whom department he visit. In public hospital this department starts working in the morning almost 7:00:00 am. In this study it is observed that there are a large number of patients who are waiting early in the morning before opening this window. For this study data is collected for only 35mints i.e .7:50:00 am to 8:25:00am. The basic queuing model observed in this department is multiple-queue multiple servers, and labeled as M/M/s, where M/M represents the Poisson probability distribution of arrivals and departures and s (positive integer) symbolize number of servers.

11.2 Out patients Department (OPD) and Queuing Model

Some specialty areas of treatment served in outpatient department are: Pediatrics, Surgery, Orthopaedics, Joint Replacement and Reconstructive surgery, Cardiology, Dermatology, Chest, Medicine, Eye, Dentistry, ENT, Obstetrics and Gynaecology, Neurosciences, Psychiatry, Pre-anesthetic and Pain clinic and Physiotherapy. Dispensaries also provide outpatient department services. Medicine outpatient department is considered for this study. In this public hospitals medicine OPD starts working from 8:00:00am to 14:00:00pm during weekdays. In this model, we have taken observations only for two hours from 9:00:00am to 11:00:00am. The basic queuing model observed in this department with queuing process is single-queue with multiple-servers, and labeled as M/M/s, where M/M represents the Poisson probability distribution of arrivals and departures and s (positive integer) symbolize number of servers and under study department observed servers were 2 i.e.s=2s.

12. CONCLUSIONS:

Queues are big problem in public hospitals. Patients have to wait long times under some times undesirable conditions. This paper has surveyed the using queuing theory for the analysis of different types of hospitals. A single server is not effective when arrival rate exceeds service rate. We can use multiple servers. An admission guarantee should be one of the main goals of any hospital for patients entering through its Emergency and Accident Department. In front hospitals many patients wait in lines and there are stress and unnecessary waiting. It provides beneficial results to be used in hospitals to make performance measures based on less waiting and less patients on queues. To reduce the average waiting time we can use current online appointment system is kind of new inventions for patients, but the unorganized rechecking and new patients without online appointment create problems in queues. They are to be separated to different lines. Many hospital staffs try to get prior treatment for their friends or relatives. These kinds of misuses are not to be allowed and seen by other patients. It is clear that there should be some penalties to hospital management and staff for not providing enough services to patients.

13. REFERENCES:

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[9]. Fatma Poni Mardiah, Mursyid Hasan Basri, The Analysis of appointment system to reduce Outpatient waiting time at Indonesia’s public hospital, Human Resource Management Research-2013,3(1):27-33.

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Received on 31.08.2016 Modified on 18.10.2016

Research J. Pharm. and Tech 2016; 9(12):2211-2216.

DOI: 10.5958/0974-360X.2016.00447.9