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THEORETICAL ARTICLE An analytical study of queues in medical sector Rashmi Agarwal1 • B. K. Singh2 Accepted: 22 August 2017 � Operational Research Society of India 2017 Abstract In general, we do not like to wait but each of us has spent a great deal of time waiting in lines even in medical sector also and queuing has become a symbol of inefficiency of a hospital. Managing the length of the line is one of the challenges facing most hospitals. A few of factors that are responsible for long waiting lines or delays in providing services are lack of passion of hospital staff and overloading of available doctors as they are attached in more than one clinic etc. This paper is based on the consideration that most of these problems can be managed by using queuing model to measure the waiting line performances as average arrival rate of patients, average service rate of patients, system utilization etc. The purpose of this study is to provide insight of general background of queuing theory, and how queuing theory can be used by policy makers to increase efficiency of services and to improve the quality of care of patients in hospitals, also understanding cost factor for getting optimum profit. Keywords Queuing theory � Priority queues � Waiting time � Performance measures and cost factor of queuing system & Rashmi Agarwal agarwal.rashmi.rashmi8@gmail.com B. K. Singh dbks68@yahoo.com 1 DAV Inter College, Bulandshahr, U.P, India 2 Head of Department of Mathematics, School of Engineering and Technology, IFTM University, Moradabad, U.P, India 123 OPSEARCH DOI 10.1007/s12597-017-0324-7 http://orcid.org/0000-0002-5246-6676 http://crossmark.crossref.org/dialog/?doi=10.1007/s12597-017-0324-7&domain=pdf http://crossmark.crossref.org/dialog/?doi=10.1007/s12597-017-0324-7&domain=pdf 1 Introduction The existence of any nation is a function of the survival of its citizens and sufficient healthcare programs of its citizenry. Hence several health policies and laws are formulated and adopted by the government. Despite of these efforts, it should be noted that, still there are some problems which challenge the success of the healthcare programs applied in medical sector. One of the most common of them is the problem of waiting lines i.e. queues found in hospitals. In general, queues form when the demand for service exceeds its supply. In medical sector, the effect of queuing in relation to the time spent by patients for treatment is increasingly concerned with modern society. Keeping patients waiting and their time wasted in the queues in hospitals often have severe consequences. Hence there is need for a significant evaluation of patient waiting time as well as reducing or eliminating it and for that queuing theory can be applied. 2 Queuing system To describe a queuing system, an input process and an output process must be specified. The input process is usually called arrival process. Arrivals are called customers and in our case ‘patients’. We assume that no more than one patient can occur at a given instant and if more than one patient can occur at a given instant, we say that ‘bulk arrivals’ are allowed. To describe the output process, we usually specify a probability distribution i.e. service time distribution, which governs a customer’s service time i.e. a patient’s service time. The uncertainties involved in the service mechanism are the number of servers i.e. doctors and the number of customers i.e. patients getting served at any time and leave the service system. Networks of queues consist of more than one server arranged in series and/or parallel. The facilities given to the customers can be classified as: • Single channel facility i.e. one queue-one service station facility: This mean that there is only one queue in which the customer waits till the service point is ready to take him for servicing. Here the customers are served by one service station only i.e. only one customer can be served at a time and the standard assumption will be made that one server can completely serve a customer. The following figure shows the single channel facility: OPSEARCH 123 • One queue—several service stations in parallel: In this case, customers wait in a single queue until one of the server stations is ready to take them for servicing. Here the customers are served by several i.e. more than one, service stations. Hence more than one customer can be served at a time as all servers provide the same type of service and each of them can completely serve the customer. Such type of service facilities is illustrated in the following figure: • Multi-channel facility i.e. many queues—many service station facilities: In such a situation, there are several queues and the customer can join any one of these, because the servers are in parallel and all servers provide the same type of service and a customer need only pass through one server to complete service. The customer joins the queue of his choice in front of service facilities and may OPSEARCH 123 be served by any service facility. The parallel arrangement is shown in the following diagram: • Multi-stage channel facilities: In this case, customers require several types of services and different service stations are there, each station providing a specialised service and the customer passes through each of the service station before leaving the system. In this arrangement, the servers are in series i.e. the series arrangement consists of a sequence of a number of service facilities such that a customer must go through one facility after another in a particular sequence before the whole service is completed. Each service facility may, however, work independently of the others, having its own rules of service. The following figure illustrates the arrangement of service facilities in series: • The mixed arrangement of service facilities: This arrangement consists of service facilities arranged in series as well as in parallel. For example: in a hospital, an incoming patient may go to any of the OPD counter (service facilities in parallel), where he is checked (served) by various doctors (facilities in series) one by one. OPSEARCH 123 Then the queue discipline is specified. The ‘queue discipline’ describes the method used to determine the order in which patients are served. The most common queue discipline is the FCFS discipline (i.e. first come, first served) in which patients are served in the order of their arrival. Under the LCFS discipline (i.e. last come, first served) the most recent arrivals are the first to enter for service. The customer i.e. patient to enter for service in randomly chosen from those patients waiting for service, it is known as the SIRO discipline (i.e. service in random order). In medical sector we mostly consider ‘Priority Queuing Disciplines’. In a priority queue discipline, each patient classifies into one of several categories. Each category is then given a priority level, and within each priority level, patients enter for service on FCFS basis. There are two types of ‘Priority Models’-Pre-emptive priority (i.e. emergency) and Non-Pre-emptive priority. In a pre-emptive queuing system, a lower priority patient can be interrupted from service whenever a higher priority patient arrives. Once no higher priority patients are present, the interrupted patient returns to service. In a pre-emptive resume model, a patient’s service continues from the point at which it was interrupted. In a non-pre-emptive model, a patient’s service cannot be interrupted. After each service completion, the next patient to enter, service is chosen by given priority to lower numbered patients types. Lower numbered are given on the basis of higher priority. Another factor that has an important effect on the behaviour of queuing system is the method that patients use to determine which line to join. 2.1 System capacity The system capacity is the maximum numberof customers, both those in service and those in the queue(s), permitted in the service facility at the same time. Whenever a customer arrives at a facility that is full, the arriving customer is denied entrance to the facility. Such a customer is not allowed to wait outside the facility (since that effectively increases the capacity) but is forced to leave without OPSEARCH 123 receiving service. A system that has no limit on the number of customers permitted inside the facility has ‘infinite-capacity’; a system with a limit has ‘finite-capacity’. 2.2 Kendall’s notation We have developed enough terminology to illustrate many queuing systems. Now, we describe standard notation used in different models in queuing theory which is initially given by D.G.Kendall. Kendall’s notation for specifying a queue’s characteristics is a/b/c/d/e, where the first characteristic ‘a’ specifies the nature of the arrival process and the second characteristic ‘b’ specifies the nature of service pattern. The following abbreviations are used for the arrival and service pattern to replace the notation ‘a’ and ‘b’: M = Markovian (or Exponential) inter-arrival time or service-time distribution. D = Deterministic (or constant) inter-arrival time or service-time. G = General distribution of service time (departure), i.e. no assumption is made about the type of distribution with mean and variance. Ek = Erlang-k distribution for interarrival or service time with parameter k (i.e. if k = 1, Erlang is equivalent to exponential and if k = ?, Erlang is equivalent to deterministic). GI = General probability distribution—normal, uniform or any empirical distribution, for inter-arrival time. The third characteristic ‘c’ specifies the number of service channels (servers), the fourth characteristic ‘d’ describes the maximum number of customers allowed in the system (in queue plus in service) and the fifth characteristic specifies the queue- discipline: FCFS = First come, first served LCFS = Last come, first served SIRO = Service in random order GD = General queue discipline. • In general, arrivals in a queuing system do not occur at regular intervals, but tend to be scattered in some manner. The Poisson assumption specifies the behaviour of arrivals, by postulating the existence of constant k, which is independent of time, queue length, or any other random property of the queue, such that p the probability of an arrival occurs between time ‘t’ and ‘tþ ot’ð Þ ¼ kot OPSEARCH 123 If the interval qt is sufficiently small and ‘n’ be a random variable (discrete) representing the number of arrivals in some time interval of length ‘t’, then ‘n’ obeys a Poisson-distribution. The probability of ‘n’ arrivals in time ‘t’ is defined as: pn tð Þ ¼ ktð Þ n�e�kt=n!; where kt is a parameter: ð1Þ In the similar way, where ‘l’ is the mean servicing rate for a particular service station then ‘n’ the number of units in the service station could serve in time ‘t’, if the service is going on throughout ‘t’ will follow the Poisson distribution, i.e. ft nð Þ probability of ’n’ servicing in time ’t’; if there were no enforced unused time½ � ¼ ltð Þn�e�lt=n!; Case-1: when n = 0. From (1), the probability for n = 0, i.e. no arrival in time qt is given by: p0 otð Þ ¼ kotð Þ 0�e�kot=0! ¼ e�kot ¼ 1� kotþ ðk2=2!Þ otð Þ2�ðk3=3!Þ otð Þ3þ � � � ¼ 1� kotþ 0 otð Þ; where 0 (qt) denotes a measure which is of smaller order of magnitude than qt as it contains higher powers of a small quantity qt. If qt is very small then 0 (qt) = 0 Hence; p0 otð Þ ¼ 1� kot i.e. the probability of no arrival in time qt = 1 - kqt. Similarly; fot 0ð Þ ¼ 1� lot i.e. the probability of no service in time qt = 1 - lqt. Case-2: when n = 1 From (1), the probability for n = 1, i.e. one arrival in time qt is given by: p1 otð Þ ¼ kotð Þ 1�e�kot=1! ¼ kot 1� kotþ ðk2=2!Þ otð Þ2�ðk3=3!Þ otð Þ3þ � � � n o ¼ kotþ 0 otð Þ; where 0 (qt) denotes a measure which is of smaller order of magnitude than qt as it contains higher powers of a small quantity qt. OPSEARCH 123 Hence; p1 otð Þ ¼ kot: If ot is very small then 0 otð Þ ¼ 0:f g i.e. the probability of one arrival in time qt = kqt. Similarly; fot 1ð Þ ¼ lot i.e. the probability of one service in time qt = lqt. Case-3: when n = m > 1 From (1), the probability for n = m[ 1, i.e. more than one arrival in time qt is given by: Pm otð Þ ¼ kotð Þm�e�kot=m! ¼ kotð Þm=m! 1� kotþ ðk2=2!Þ otð Þ2�ðk3=3!Þ otð Þ3þ � � � n o ¼ km=m! otð Þm�k otð Þmþ1þ � � � n o ¼ 0 otð Þ; where 0(qt) denotes a measure which is of smaller order of magnitude than qt as it contains higher powers of a small quantity qt. Hence; pm otð Þ ¼ 0: If ot is very small then 0 otð Þ ¼ 0:f g i.e. the probability of more than one arrival in time qt = 0. Similarly; fot mð Þ ¼ 0 i.e. the probability of more than one service in time qt = 0. Hospitals are like service industry with different branches of services as doctors, nurses, and other paramedical services. The treatment process consists of a set of activities and procedures that the patient must go through in order to receive the required treatment. In order to represent the workflow in a hospital, it is important to understand how the departments of the hospital operate and how data flows. The following flow- diagram shows the hospital system: OPSEARCH 123 2.3 Hospital-system Now for calculations, we define k to be the ‘arrival rate’, which will have units of arrivals per minute and l to be ‘service rate’. Ws is the expected time that a patient spends in the queuing system, including time in line (i.e. queue) plus time in service, and Wq is the expected time that a patient spends waiting in line. We have assumed that the arrival rate, service rate and number of servers has stayed constant over time, this is known as steady state of the system. In many situations, the arrival rate, the service rate, and number of servers may vary over time, and then we say that the queuing system is not stationary. We are using the queuing model (M/M/1): (?/FCFS) which is also known as Birth and Death Model as here is Poisson arrival and Poisson service. We are OPSEARCH 123 assuming that servicing is through single channel with infinite capacity (because mostly hospitals give 24-h service). The service discipline is first come first serve. Case-1: there is no patient in the queue in a hospital at time (t 1 qt) This follows in two ways: 1. The probability of no patient in the queue in a hospital at time ’t’ = p0 (t) No patient arrives in time qt, with probability p0 (qt) = 1 - kqt. Hence, the probability in this case is p0 (t)�(1 - kqt). 2. There is one patient in the queue in a hospital at time ‘t’, with probability p1 (t) No patient arrives in time qt, with probability p0 (qt) = 1 - kqt. One service in time qt, with probability fqt (1) = lqt. Hence, the probability in this case is p1 (t)�(1 - kqt)�lqt. Since above two ways are mutually exclusive, therefore, p0 (t ? qt), the probability of no patient in the queue in a hospital at time (t ? qt) is obtained by adding the above two probabilities. i:e:; p0 tþ otð Þ ¼ p0 tð Þ � 1� kotð Þ þ p1 tð Þ � 1� kotð Þ � lot or; p0 tþ otð Þ ¼ p0 tð Þ þ �kp0 tð Þ þ lp1 tð Þ½ � � otþ �klð Þ � p1 tð Þ � otð Þ 2 or; p0 tþ otð Þ ¼ p0 tð Þ þ �kp0 tð Þ þ lp1 tð Þ½ � � otþ 0 otð Þ or; p0 tþ otð Þ � p0 tð Þ½ �=ot ¼ �kp0 tð Þ þ lp1 tð Þ½ � þ 0 otð Þ=ot Taking limit as qt ? 0, we get d=dt p0 tð Þ ¼ �kp0 tð Þ þ lp1 tð Þ½ � ð2Þ Case-11: there are n > 0 patients in the queue in a hospital at time (t 1 qt) This occurs in the following ways— (a) There are (n - 1) patients in the queue in a hospital at time ‘t’, with probability pn-1 (t) One patient arrives in time qt, with probability p1 (qt) = kqt. No service in time qt, with probability fqt (0) = 1 - lqt. Hence, the probability in this case is pn-1(t)�kqt�(1 - lqt). (b) There are ‘n’ patients in the queue in a hospital at time ‘t’, with probability pn (t) No patient arrives intime qt, with probability p0 (qt) = 1 - kqt. No service in time qt, with probability fqt (0) = 1 - lqt. Hence, the probability in this case is pn(t)�(1 - kqt).(1 - lqt). (c) There are (n ? 1) patients in the queue in a hospital at time ‘t’, with probability pn?1(t) No patient arrives in time qt, with probability p0 (qt) = 1 - kqt. One service in time qt, with probability fqt (1) = lqt. Hence, the probability in this case is pn(t)�(1 - kqt)�(lqt). OPSEARCH 123 Since above ways are mutually exclusive, therefore, p0 (t ? qt), the probability of ‘n’ patients in the queue in a hospital at time (t ? qt) is obtained by adding the above probabilities. i:e:; pn tþ otð Þ ¼ pn�1 tð Þ � kot � 1� lotð Þ þ pn tð Þ � 1� kotð Þ � 1� lotð Þ þ pnþ1 tð Þ � 1� kotð Þ � lot or; pn tþ otð Þ ¼ kpn�1 tð Þ� kþ lð Þpn tð Þ þ lpnþ1 tð Þ � � � otþ pn tð Þ þ 0 otð Þ or; pn tþ otð Þ � pn tð Þ½ �=ot ¼ kpn�1 tð Þ� kþ lð Þpn tð Þ þ lpnþ1 tð Þ � � þ 0 otð Þ=ot Taking limit as qt ? 0, we get d=dt pn tð Þ ¼ kpn�1 tð Þ� kþ lð Þpn tð Þ þ lpnþ1 tð Þ � � ð3Þ Now when the system reaches steady state, lim t!1 pn tð Þ ¼ pn and lim t!1 d=dt pn tð Þ ¼ 0 Therefore, under steady state of the system, the Eq. (2) reduces to �kp0 þ lp1 ¼ 0 Or; p1 ¼ k=lð Þp0; where k=l\1ð Þ ð4Þ And the Eq. (3) reduces to— kpn�1� kþ lð Þpn þ lpnþ1 ¼ 0 Putting n = 1, 2… etc. in the above equation, we get – p2 ¼ �k=lð Þp0 þ k=lþ 1ð Þp1 ¼ �p1 þ k=lð Þp1 þ p1; from 4ð Þf g ¼ k=lð Þp1 ¼ k=lð Þ2p0 and p3 ¼ k=lð Þ 3 p0; . . .and so on: Proceeding in this way, we have— pn ¼ k=lð Þ n p0; for n� 0: But X1 n¼0 pn ¼ 1 OPSEARCH 123 or; p0 þ p1 þ p2 þ p3 þ � � � ¼ 1 or; 1þ k=lð Þ þ k=lð Þ2þ k=lð Þ3þ � � � h i p0 ¼ 1 or; 1= 1�k=lð Þ½ �p0 ¼ 1 wherek=l\1ð Þ or; p0 ¼ 1� k=lð Þ Therefore, the probability of ‘n’ patients in the queue in a hospital at any time pn = (k/l) n[1 - (k/l)]. 2.3.1 Cost factor of the queuing system In today’s throat cut competition for all healthcare organizations, they always look for more and more customers for generating maximum profit and providing the best available service to the patients. Also the large segments of our organizations operate in such a way that consumers have no real capability to learn about price and quality. We shall discuss mainly two types of costs associated with the service providing activities, they are: • The cost of waiting in line The problem in almost every queuing situation is traffic and the supervisor must think about the additional cost of providing more rapid services such as more checkouts, more attending staff etc. against the intern cost of waiting. For example, if patients are walking away disgusted because of insufficient customer support persons, the business could compare the cost of hiring more staff to the value of increase of revenues and maintaining customer loyalty. Waiting time cost consists of all the costs which incurred to the hospital caused by the dissatisfaction of the patients due to wastage of their valuable time as standing in queue for getting the service. TotalWaitingCost ¼ Number of arrivalsð Þ � Averagewait per arrivalð Þ � Cost of waitingð Þ ¼ k�Wq � Cw: • Service cost Service cost is defined as all the costs associated with service providing activities as we know that any providing service is not free of cost. These costs are directly related with the facility provided to the patients arrived for treatment. It is a substantial cost incurred for the service provider; this is the combination of both fixed as well as variable cost for any service. For example, fees of the doctor, salary of the staff, rent of the room, electricity and telephone bills, required stationary items etc. Total Service Cost ¼ Number of channelsð Þ � Cost per channelð Þ ¼ m� Cs: The person who is responsible for the management of all the activities in a hospital has to analyze these two types of costs as both of them are inverse in nature, OPSEARCH 123 on reducing one type of the above mentioned two costs the other cost increases and vice-versa. Initially, the cost of waiting in line is at the maximum, when the hospital is at the minimum service capacity. As service capacity increases, there is a reduction in the number of patients in the line and in their wait times, which decreases queuing cost. The relationship between these can be expressed graphically as follows: The optimal total cost is found at the intersection between service capacity and waiting line curves. The upper curved line in the graph represents the total cost associated with the service providing operations. Total Cost ¼ TotalWaiting TimeCost þ Total Service Cost 3 Review of literature The application of queuing theory within the medical sector has been in existence for over 50 years. Waiting time is considered to be an important part in a hospital system. Now an overview of previous work falling within the scope of the paper is presented. Gupta et al. [1], Adeleke et al. [2], and Brohma [3] planned man power as how large the staff is required to give adequate service. According to them more doctors should be deployed to convert the single-channel queuing units to multi-channel units. Marvasti [4] and Mishra [5] were used survey methodology to avoid delays at medical processes and compute the total optimum cost of independent queuing systems with controllable arrival rates. Fomundan and Herrmann [6], Wang et al. [7], Fazlul et al. [8], Obamiro [9], Foaster et al. [10], Soni and Sexena [11] and Mehendiratta [12] proposed to use queuing theory models in healthcare process and developed some new methods. Barlow [13] makes some recommendations for removing the waiting problems. OPSEARCH 123 Verma [14] evaluates all the costs, profits and utilization factor for an optimum profit. 4 Objectives of the study The objectives of this work are as follows: • To obtain probability of patients in a queue in a small time interval according to gender of respondents. • To compare waiting time of patients in the system according to gender of respondents. • To compare waiting time of patients in the queue according to gender of respondents. • To determine the length of queue according to gender of respondents. • To obtain probability of patients in a queue in a small time interval on the basis of age-groups of respondents. • To compare waiting time of patients in the system on the basis of age-groups of respondents. • To compare waiting time of patients in the queue on the basis of age-groups of respondents. • To determine the length of queue on the basis of age-groups of respondents. • To obtain the cost of the queuing system based on time spends in the queue according to gender of respondents. • To obtain the cost of the queuing system based on time spends in the queue on the basis of age-groups of respondents. 5 Methodology The admitting department is one of the most congested hospital services, and faces a great deal of pressure compared with other components of the hospital system. In this study, field observations are conducted to determine the current operation of the admitting department. The author collects actual data for arrivals, waiting times and service times. The data is collected from 60 peoples of Meerut through the questionnaire. The actual workload within the hospital is difficult to measure due to its multifunctional systems; therefore, in order to obtain a reliable figure, the data is collected for different factors related to a hospital system. This data will enable us to obtain the arrival rate, the service rate and performance measures regarding queue lengths. OPSEARCH 123 6 Work The data collected by the questionnaire with the sample size 60 is analysed as under: 0 10 20 30 40 50 60 70 80 90 100 MALE FEMALE N O . O F RE SP O N DE N TS → OPTIONS → Sex of Respondents 56.7% 43.3 % Table 1 and the above graph show that out of 60 (100%) respondents, 34 (56.7%) respondents were male and 26 (43.3%)were female and the findings are shown in the following table according to gender of respondents: S.no. Performance measures Male respondents Female respondents 1. Mean arrival rate of patients [k (approx. value)] 14 15 2. Mean service rate of patients [l (approx. value)] 15 16 3. Average time of patients spend in the system [Ws = 1/ (l - k)] 60 s 60 s 4. Average time of patients spend in the queue [Wq = (k/l)(1/ l - k)] 56 s 57 s 5. Average length of queue = [l/(l - k)] 15 16 By calculation, the author finds that by the opinion of male respondents, the average time of patients spend in the system is 60 s and that according to female respondents also, it is 60 s. The average time of the patients spend in the queue is approximately 56 s by the opinion of male respondents and that according to female respondents is 57 s while the average length of queue according to male respondents is 15 patients and by the opinion of female respondents, it is 16 patients. Hence the author finds that there is a little difference between the opinions of male and female respondents. Table 1 Distribution of sample according to sex Sex No. of respondents Percentage Male 34 56.7 Female 26 43.3 Total 60 100 OPSEARCH 123 6.1 To obtain the probabilities of patients in the queue in a hospital according to gender As we know that, p0 = 1 - (k/l), p1 = (k/l) p0, p2 = ((k/l) 2 p0 (where k/l\ 1) The following table shows their values: Probabilities Male Female p0 0.067 0.062 p1 0.062 0.058 p2 0.058 0.055 The above table shows that the probability of no male in the queue is 0.067 and that of female is 0.062. The probability of one male in the queue is 0.062 and that of female is 0.058, also the probability of two males in the queue is 0.058 and that of female is 0.055. Hence, we find that the probability of females is less than that of males for waiting in the queue, also probability of waiting in the queue decreases as the number of patient increases in the queue, this may be depend upon speed of the service. Table 2 and the above graph show that out of 60 (100%) respondents, the age of 8 (13.3%) respondents were below 18 years, 34 (56.7%) were within the range of 18–35 years, 13 (21.7%) were within the range of 35–50 years and 5 (8.3%) were above 50 years. After the calculations for performance measures, the findings on the basis of age-groups of respondents are shown in the following table: OPSEARCH 123 S.no. Performance measures Results according to age-group of respondents Below 18 years 18–35 years 35–50 years Above 50 years 1. Mean arrival rate of patients [k (approx. value)] 13 15 16 11 2. Mean service rate of patients [l (approx value)] 14 16 17 13 3. Average time of patients spend in the system [Ws = 1/(l - k)] 60 s 60 s 60 s 30 s 4. Average time of patients spend in the queue [Wq = (k/l)(1/l - k)] 56 s 56.25 s 56.5 s 25.4 s 5. Average length of queue = [l/(l - k)] 14 16 17 7 (approx.) Hence the author finds that the average time of the patient spend in the system is 60 s by the opinion of respondents of the age-groups below 18 years, 18–35 years and 35–50 years while according to the respondents of age above 50 years, it is 30 s. Average time of the patient spend in the queue is approximately 56 s while the average length of queue is 14 patients according to the respondents of the age below 18 years. According to the respondents of age-group 18–35 years and 35–50 years, average time of the patient spend in the queue is 56.25 and 56.5 s respectively and average length of queue is 16 and 17 patients respectively, but the opinion of the respondents of the age above 50 years is that the average time of the patient spend in the queue is 25.4 s and average length of queue is 7 patients. 6.2 To obtain the probabilities of patients in the queue in a hospital according to the age-groups Probabilities Below 18 years 18–35 years 35–50 years Above 50 years p0 0.071 0.062 0.059 0.154 p1 0.066 0.058 0.056 0.130 p2 0.061 0.055 0.052 0.110 Table 2 Distribution of sample according to age Age No. of respondents Percentage Below 18 years 08 13.3 18–35 years 34 56.7 35–50 years 13 21.7 Above 50 years 05 8.3 Total 60 100 OPSEARCH 123 The above table shows that the probability of no patient in the queue for the patient below 18 years, 18–35 years, 35–50 years and above 50 years are 0.071, 0.062, 0.059 and 0.154 respectively. The probability of one patient in the queue in a hospital for the patient below 18 years, 18–35 years, 35–50 years and above 50 years are 0.066, 0.058, 0.056 and 0.130 respectively, also the probability of two patients in the queue for the patient below 18 years, 18–35 years, 35–50 years and above 50 years are 0.061, 0.055, 0.052 and 0.110 respectively. Hence, we find that the probability of patients waiting in the queue decreases as age of the patient increases but for old patients this increases, also probability of waiting in the queue decreases as the number of patient increases in the queue as we see the same in the case on the basis of gender. The probability for old patients is high; hence this should be reduced by given special facilities to them in a hospital so that they do not suffer and get the treatment in time. From the above comparisons based on gender and different age-groups of the respondents, the author finds that the average waiting time is not much but if this became negligible by the application of queuing theory then the average length of queue can be reduced and the system will be an ideal system, also there requires to pay attention on old patients in a hospital so that they could get treatment in time. 6.3 To obtain the cost of the queuing system based on time spends in the queue For this, we assume that the hospital’s facility can handle only one emergency at a time as we are using (M/M/1):(?/FCFS) queuing model. Suppose that it costs the hospital Rs. 100 per minute and each minute of waiting costs by Rs. 10. As we know that, Total waiting cost ¼ k�Wq � Cw and Total service cost ¼ m � Cs The following tables show their values: According to gender of respondents: Costs (in Rs.) Male Female Waiting cost 130.70 142.50 Service cost 100.00 100.00 Total cost 230.70 242.50 On the basis of age-groups of respondents: Costs (in Rs.) Below 18 years 18–35 years 35–50 years Above 50 years Waiting cost 121.30 140.60 150.70 46–60 Service cost 100.00 100.00 100.00 100.00 Total cost 221.30 240.60 250.70 146.60 OPSEARCH 123 A final approach is evaluation of ‘‘cost of illness’’ which includes ‘direct costs’ (where money actually charges), ‘indirect costs’(the value of lost output from time off work due to illness) and ‘intangible costs’(the disvalue to an individual of pain and suffering). The optimal level of investment in health occurs where the ‘marginal cost’ of health capital is equal to the ‘marginal benefit’. 7 Conclusion and recommendations Unnecessary waste of time in the hospitals or healthcare centres may lead to patients’ health complications and in some cases eventual death which may be avoided. There are several probable ways of improving patient flow, and thus reducing waiting time for the patients, such as: • By increasing the number of servers, • By controlling the arrival rate; and • By optimizing the service rate, etc. The number of servers can be increased by hiring more admitting staff which will present an immediate improvement in services of registration system. Pre-registering a large number of patients and visiting the staff to serve directly the patient will eliminate the crowd in the visiting-room.More doctors and more paramedical officers should be deployed to healthcare centres for taking patients’ preliminary tests or service before they spot the doctor. This will reduce the service time of the doctor in attending to patients and thus increase the service efficiency. 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SSRG-IJME 2(5), 43–49 (2015) OPSEARCH 123 http://dx.doi.org/10.1186/1471-2288-10-60 An analytical study of queues in medical sector Abstract Introduction Queuing system System capacity Kendall’s notation Hospital-system Cost factor of the queuing system Review of literature Objectives of the study Methodology Work To obtain the probabilities of patients in the queue in a hospital according to gender To obtain the probabilities of patients in the queue in a hospital according to the age-groups To obtain the cost of the queuing system based on time spends in the queue Conclusion and recommendations References
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