QUEUING THEORY 4.1 INTRODUCTION

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1 C h a p t e r QUEUING THEORY 4.1 INTRODUCTION Queuing theory, which deals with the study of queues or waiting lines, is one of the most important areas of operation management. In organizations or in personal life, there are examples of processes which generate waiting lines or queues. Such waiting lines occur because the current service facility insufficient to provide service at that instance. On traveling by airlines, we have first hand experience with several types of waiting lines (queues). To buy ticket, we may have to stand in line at travel agent's office. When we arrive at airport we stand in line to check baggage, then we stand in line again to get a seat assignment. We line up once more for a security check and then again in the boarding lounge before entering the airplane. When we are inside the plane, we wait for those ahead of us to take their seats. The plane itself waits for take off clearance, when it arrives at its destination, it may circle for some time waiting for landing clearance. And finally, we may wait for baggage to arrive and then for ground transportation. It's possible to be a member of at least 10 queues on one such trip. Other instances where the queues are formed are: 1. Computer programme are waiting to be processed at computer center. 2. Customers are waiting to be served at as bank teller's window. 3. Parts are waiting to be processed at a manufacturing operation. 4. Machines are waiting to be repaired at a maintenance shop. 5. Trucks are waiting to unload their cargo at unloading dock. A queue is formed under two conditions. The first condition is "Customers wait for getting service". This means that the service facility is not having appropriate capacity to render the service as and when customers arrive and the customers have to wait or stand in queue for getting service later on. More precisely this is the condition where arrival rate of customers at the service facilities are higher than the service rate.under second condition "Service facilities wait for customers". This means that the service facilities are having excessive capacity and are staying idle due to lack of customers. More precisely this conditions can be stated as "service rate of facilities are higher than the customers arrival rate ". The central problem in virtually every waiting line situation is a trade-off decision between cost of service capacity and waiting line cost. Providing too much service facilities for eliminating customer queue involve excessive cost that might not be justifiable. Since unlimited capacity could not be installed and capacity required significantly differ from peak or rush hours compared to normal hours, it is quite difficult to fix capacity requirement. On the other hand, not providing enough service capacity would cause the-waiting lines to become excessively long at times. Customers even before joining the queue get discouraged by seeing the excessive waiting time for service or customers after joining the queue leave the service system due to intolerable delay. Therefore the ultimate goal is to achieve the economic balance between cost of service capacity and waiting line cost. Queuing theory contributes vital information required for such a decision by predicting various characteristics of waiting line such as the average number of customer in queue or system, time spent by customer in queue or system etc. Based on the probability theory it attempts to minimize the extent and duration of queue with minimum of investment cost. This credit of analyzing queues goes to Danish Engineers, A.K. Erlang; in his attempts eliminate bottlenecks created by telephone calls on switching circuits. This queuing theory can be applied to a wide variety of

2 Queuing Theory operational situations where imperfect matching between customer and service facilities is caused by one's inability to predict accurately the arrival and service time of customer. Queuing models calculations is sometime longer and more tedious. Various easier software for handing queuing problems have been already available DEFINITION AND OBJECTIVES OF QUEUING THEORY A queue (or a waiting line) is a line or list of customers who remain waiting for getting certain goods or service from service center. There are two types of line formed namely a physical line or a list of dispersed customers. When the customers remain standing in a line in front of a service center or counter, it is called physical queue of waiting line. When the customers remain waiting for certain service after enlisting their names or submitting their application in the service center, it is called waiting list of customers, e.g. list of customers waiting for gas cylinders, admission into an institution etc. The queuing (waiting line) theory deals with the analysis of queues and queuing behavior by finding solution to the problem relating to the optimization of the effectiveness of the defined function with random times of arrival and service. Optimization here referred to minimization the total cost of queuing system. Total cost include : Cost of providing service Cost of waiting These two costs are inversely proportional to each other. Addition of more service centers will increase the service capacity and ultimately minimize the waiting line or waiting cost. However addition of service facility would ultimately increase the costs providing service. Similarly, if the numbers of service centre are reduced, it minimizes the service costs but increase the waiting cost. Thus the objective of queuing theory is to minimum the total cost associated with the queuing system. The relationship between these two costs is shown in figure 4.1. Figure 4.1 : Cost Trade Off of operating service facility Optimal cost of system Total cost of system Total expected cost of operating facility Cost of service capacity Waiting line cost Increased service Optimal service level It is clear from the figure that an increase in the service facility (service center) will reduce the waiting cost and increase the cost of providing service. The total cost, which is obtained by adding these two costs component together decreases up to a point and then rises as more and more service is provided. Thus the objective of the queuing theory is simply to determine the service level where the total cost of system is lowest. This is the point where cost of service capacity line and waiting line cost cross each other At this point of minimum total cost, waiting line cost will be equal to cost of providing service. However, it is important to note that queuing theory does not directly solve the problem of minimizing the total waiting and service cost but the theory provides the management with necessary information by estimating different characteristics of the waiting line such as: (a) Average arrival rate (b) Average service rate (c) Average length of queue (d) Average waiting time (e) Average time spent in system 4.3. PROBLEM INVOLVING QUEUING THEORY Queuing theory can be successfully used in various problems where the total cost of queuing system can be minimized by reducing the idle time of service center and minimizing the customers total waiting time. Some general types of queuing situations which have economic consequences are: 57

3 A Text Book of Operational Research and Food Plant Management 1. Industrial production process a. Facilities required keeping a batch of machine in economic operation.. b. Supply of raw materials and dispatch of finished products. c. Costly items in stock (inventory). d. Assembly lines. e. Tool room service. f. Bills in A/c department-payments/outstanding g. Storage /dumps h. Computed service 2. Transport: a. Number of bus terminals/bus stops b. Number of siding/platforms. c. Number of runways/checking counters in airports. d. Shipping. 3. Communication: a. Trunk calls - Number of booths/lines b. Telephones - Number of booths/lines 4. Public service industry a. Hospital wards b. Out patient department required in hospital c. Level crossings /Tool booths required / Ticket counted d. Bank / insurance company 5. Others a. Human relations/ Co-ordination - Numbered of subordinates to an executive. b. Waiting to promotions c. Theatres / hours for arranging screening of pictures / wedding / meetings. d. Waiting for tickets to see pictures QUEUING SYSTEM AND ITS ESSENTIAL ELEMENTS The essential features of a queuing system are shown is figure 1.2. This consists of: Input source (or calling population) Queuing process Queue discipline Service process (or mechanism) Departure pattern Customers requiring service are generated at different times by an input source, commonly known as population. The rate (constant or random) at which customers arrive at service facility is determined by the arrival process. Customers' entry into the service system depends upon the queue conditions. Customers are served immediately if the service facility is idle at the time of their arrival. However if the facilities are not idle they have to stand on the queue, which can have the several configurations. Figure 4.2 : Queuing System Service system Waiting customers Input Source Arrival process Queuing Process Queuing discipline Service Process Departure (Served customers) Balk Renege Jockey Customers from the queue are selected for service according to certain rule known as queue discipline. The service facility may consists of no server (self service), one or more server arranged in series of parallel. The rate 58

4 Queuing Theory (constant of random) at which customers arrive and service is rendered is known as arrival rate and service rate respectively. Once the service is rendered customers leave the system which is called as departure INPUT SOURCE The input sources are the finite or infinite sources of potential customers commonly known as populations that will use the service. Input source has the following three characteristics: Size of calling population, Behavior of the arrivals, and Pattern of arrival of customers at the system Size of calling population: Arrival at a service system may be drawn from a finite or an infinite population. A finite population refers to a limited size customer pool that will use the service and, at times, form a line. In finite population when a customer is getting service, the size of population is decreased and the probability of user requiring service is also decreased. Conversely, when a customer is serviced and returns to the user group, the population increases and the probability of a user requiring service is also increases. As an example, consider a group of six machine (finite population) maintained by one repairperson. When one machine breaks down, the source population is reduced to five, and the chance of on of the remaining five breaking down and needing repair is certainly less than when six machines were operating. If two machines are down with only four operating, the probability of another breakdown is again changed. Conversely when the machine is repaired and returned to service, the machine population increases, thus raising the probability of next breakdown An infinite population is large enough in relation to the service system so that the population size caused by subtractions or additions to the population (a customer needing service or a serviced customer returning to the population) does not significantly affect the system probabilities. Examples of infinite population include customer arriving at a bank of super market, students arriving to get admission at the university, cars arriving at a highway petrol pump etc. An input source need not be homogeneous population but may consist of several sub-populations. For example, patient arriving at OPD of a hospital are normally of three categories: walk in patients, patients with appointments and emergency patients. Each patient's class places different demands on service facility, but the waiting expectations of each category differ significantly. Behaviour of Arrivals: The arriving customer shows different kind of behaviors. Some customer, on arriving at the service system stays in the system until served, no matter how much he has to wait for service. This type of customer is called a patient customer. Machines arrived at the maintenance shop in a plant are examples of patient customers. Whereas the customers, who waits for a certain time in the queue and leaves the service system without getting service due to certain reason such as long queue in front of him is called an impatient customer. There are two classes of impatient arrival. Members of the first class arrive, survey both the service facility and the length of the line, and then decide to leave. Those in the second class arrive, view the situation, join the waiting line and after some period of time depart. The behavior of first type is termed as balking and the second is called reneging. A third behavior is often seen where customers move from one queue to another hoping to receive service more quickly. This type of behavior is termed as Jockeying. Pattern of Arrivals: The arrivals patterns behavior of customer to the service system is either static or dynamic. In the statistic arrival process, control is achieved by customer only and the service facility remains of fixed capacity. The static arrival processes are either in constant fashion or on random fashion. For example in clinic, patients may be appointment in such a manner that they arrive at the clinic at specific equal interval of time. On other hand, the arrivals times of customers on a restaurant are of random fashion and can not be predicted. The dynamic arrival process is controlled by both service facility and customers. The service facility is adjusted for its capacity to match changes in demand intensity either by varying staff levels or varying service charge (such as telephone has different charge at different times). The variation in demand intensity also affects the customer behavior. They either balked or renege from the service system when confronted with a long or slow money waiting lines. 59

5 A Text Book of Operational Research and Food Plant Management Waiting line formulas generally require an arrival rate i.e. the number of units per period such as average of one customer in every five minutes. When arrivals at a service facility occur in a purely random fashion, a plot of inter-arrival times yields an exponential distribution. On the other hand the number of arrivals during some time period "T" follows Poisson distribution. This can be summarized that time between arrivals is exponentially distributed and the number of arrivals per unit time is Poisson distributed. The commonly used symbol for mean arrival rate in queuing model is. (Greek letter, lamda), thus the time between successive arrivals can be expressed as 1/ QUEUING PROCESS The queuing process refers to the number of queues (lines) and their respective lengths. The number of queues may be single one or multiple depend upon the layout of a service system and the type of service to be rendered. Regarding length of queue it might be finite length or infinite length. Whenever a service system is unable to accommodate more than the specified number of customers at a time and no further customer are allowed to enter until space becomes available to accommodate new customers are called finite queues. For example gas stations, loading docks, parking lots, cinema halls, restaurants etc. have a limited line capacity caused by legal restriction of physical space characteristics. In practical sense, an infinite line is simply one that is very long in terms of the capacity of the service system however in queuing model, if a service system is able to accommodate any number of customers at a time, then it is referred as an infinite or unlimited source queue e.g. customers order receive at sales department QUEUE DISCIPLINE A queue discipline is a priority rule or set of rules for determining the order of service to customers in a waiting line. The rules selected can have dramatic effect on the system's overall performance. Number of customer in line, the average waiting time, range of variability in waiting time and the efficiency of the service facility are few factors affected by the choice of priority rules. Some of the major queue disciplines are: 1. First-come, first served (LCFS): Customers are served in the order of their arrival. This type of discipline is found in ticket counter, bills paying at bank, telephone counter, ration shop etc. 2. Last-come, first-served (LCFS): Under LCFS, customer or unit coming last are served first. This discipline is practiced in most cargo handling situations where the last item loaded is removed first. Similarly, in work shops, items in the top of stock is taken first for processing, which is the last one to arrive for service. 3. Service in random order (SIRO): Under this discipline customers are selected for service at random irrespective of their arrivals in service system. 4. Priority Service: Under this rule, customers are grouped in priority classes on the basis of some attributes such as service time or urgency, and FCFS rule is used within each class to provide service. 5. Pre-emptive priority: Under this rule, the highest priority customer is allowed to enter into the service immediately after entering into the system even if a customer with lower priorities is already in service. Thus lower priorities customers' service is interrupted (pre-empted) due to arrival of highest priority customer. For example serious patient is given highest priority in hospital irrespective of his arrival time. 6. Non-pre-emptive priority: In this case highest priority customer goes ahead in the queue, but service is started immediately on completion of the current service SERVICE PROCESS (OR MECHANISM) The service process is concerned with the manner in which customers are serviced and leave the service system. It is characterized by: The arrangement and capacity of service facility The distribution of service times. Arrangement and Capacity of Service Facility: The capacity of service mechanism is measured in terms of customers that can be served simultaneously and effectively on unit time e.g. hour, day etc. The arrangement of service facilities may have different layouts such as series arrangements, parallel arrangements or mixed (partly in series and partly in parallel) arrangements. 60

6 Queuing Theory Series Arrangements: 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 rule of service. For example, during university / college admissions, the student go through one counter after another before their admission formalities is completed.the series arrangement can have single queue, single server or single queue multiple servers as shown in figure 10.3 and 10.4 Figure 10.3: Single queues, single server Customers Service facility Serviced Customer Figure 10.4 single queues, multiple server Customers Service facilities Serviced Customer Parallel Arrangements In parallel arrangement, the service facilities are arranged in parallel to each other. This provides opportunity for customer to join any service center of his choice. Check in points in airports is the examples of parallel arrangement of service facilities. Pictorial representation of parallel arrangement is shown in figure 4.5. Figure 4.5: Parallel arrangement of service Facilities Customers Service facilities Served customers Mixed Arrangements Mixed arrangement consists of service facilities in both series as well as parallel form. These types of arrangement are often seen in hospitals or universities. Figure 4.6 : Mixed arrangements of service facilities Customer Service facilities Served customers Distribution of service time: The time interval from the commencement of service to the completion of service for a customer is known as service time. It is generally expressed as a unit time e.g. average five minutes per customers. The capacity of 61

7 A Text Book of Operational Research and Food Plant Management servers in number of units per time period is called service rate. The service time may be either constant or scattered in some fashion for different customers. The service time distribution can be described in terms of either Poisson or as Exponential distribution. However exponential distribution of service time is commonly used for mathematical purpose on queuing model. The commonly used symbol for the mean service rate is (Greek Letter pronounced as 'mu') and thus the mean service time or average time between services is 1 / Departure Pattern Generally this factor can be ignored but some time it may influence service and/or arrival times. For example, if there is only one door to service point through which people enter and leave after being served, it is possible that people leaving could affect the rate of arrival. In a single channel facility, out put of the queue does not pose any problem for the customer that departs after receiving the service. However, output of the queues become important when the system is of multistage channel facilities because the possibility of a service station breaking down can have repercussions on the queues SINGLE CHANNEL QUEUING MODEL In this section, we introduce an analytical approach to determine important measure of performance in a typical single channel service system ASSUMPTION OF SINGLE CHANNEL QUEUING MODEL The single channel, single phase model considered have is one of the most widely used and simplest queuing models. It assumes the existence of following seven conditions. (1) Arrivals are served on the first come first served (FCFS) basis. (2) Every arrivals waits to be served regardless of the length of line (i.e., there is no balking or reneging). (3) Arrivals are independent of preceding arrivals, but the average number of arrivals (the arrival rate) does not change over time. (4) Arrivals are described by a Poisson probability distribution and come from an infinite or very large population. (5) Service time also varies from one customer to next and is independent of the, but their average rate is known. (6) Service time occur according to the negative exponential probability distribution. (7) The average service rate () is greater than the average arrival rate () i.e. ratio / which is called traffic intensity or utilization factor is always less than one, and the length of queue will go on diminishing gradually OPERATING CHARACTERISTICS OF A QUEUING SYSTEM Queuing models enables the analyst to study the effects of manipulating decision valuables on the operating characteristics of a service system. Decision variables are related to arrival rate, number of service facilities, number of phases, number of server per facility, priority discipline, queue arrangement etc. Some of the more commons operating characteristic are as follows: 1. Queue length: It is the number of customer in waiting line. It can be either short queue or long queue. If queue is short it is assumed that either service facilities are having excessive service capacity or servers are providing efficient service. A long queue indicates either low service efficiency of server or inadequate capacity of service facilities. 2. Number of customer in system (System length): It is the number of customer waiting in queue and being served. System length provides information about service efficiency and capacity. Large values of service length may generate potential customer dissatisfaction and to avoid this, there is a need for increasing service capacity. Conversely, a small value of system length indicates excessive capacity of facilities or highest efficiency of service station. 3. Waiting time in queue: It is the average time spent by a customer in the queue before the commencement of his service and can be used to evaluate the quality of service. Long lines do not reflects the long waiting times if the service rate if fast. However, when waiting time seems long to customers, they perceive that the quality of service is poor. Long waiting times may indicate a need to adjust the service rate of the system or change the arrival rate and pattern of customers. 4. Waiting time in system: It is the average time spent by a customer in a system. It includes time spent by a customer in queue and service center i.e. waiting time plus service time. The total elapsed time from entry into 62

8 Queuing Theory the system until exit from the system may indicate customer satisfaction level, server efficiency, and server capacity. It the customers are spending too much time in the service system, there may be a need to change the priority discipline, increase efficiency or adjust capacity in some way. 5. Service facility utilization: It is a proportion of time that a server actually spends with a customer. It gives an idea of the expected amount of idle time which can be used for some other work not directly involved with service. Management is interested in maintaining high utilization but objective may adversely impact the other operating characteristics. The other operating characteristics are probability that the service facility remains idle and the probability of specific number of customers or units in the system NOTATIONS USED IN SINGLE CHANNEL QUEUING SYSTEM Notations used to analyze the queuing system are as follows: = Mean customer arrival rate defined as average number of arrivals in queuing system per unit time or mean number of arrivals per time period (e.g., per hour) = Mean service rate defined as average numbers of customers completing service per unit time or mean number of customers (or units) served per time period. 1/ = Mean time between arrivals. 1/ = Mean time per customer served. ρ = Traffic intensity or the server utilization factor defined as the expected fraction of for which server is busy). It is expressed as Mean time per customer served (1/) ρ = = Mean time between arrivals (1/) L s = An average number of customers (or units) is the system (i.e., the numbered in line plus number being served). W s = An average time a customer spends in the system. (i.e., the time spent in line plus the time spent being served). L q = An average number of customers in waiting line or queue (queue length). W q = An average time, a customer spends waiting in the queue. n = Number of customers is service system. P n = Probability that there is 'n' number of customers in the system QUEUING EQUATIONS Majority of quantitative methodologies used in the development of most waiting line models is rather complex and outside the scope of this text. However, the quantitative expressions that have been developed to single channel waiting lines are given below: Let be the mean or expected number of arrivals per time period (mean arrival time) and be the mean or expected number of items served per time period (mean service rate) and using the assumptions of Poisson arrivals and exponential service time distribution, the following equations could be developed. 1. Utilization factor or traffic intensity defines as probability that service facility is being used. ρ = = Average rate of arrival / Average rate of service 2. The probability that service facility is idle (i.e., the probability of no units in the system). P 0 = 1 - = 1-ρ 3. The probability that there we is 'n' units in the system (units in the queue or waiting line plus numbered being served ). P n = P 0 or, P n = 1 - n = P 0 (ρ) n = (1-ρ) (ρ) n n 4. Expected or mean number of customer (units) in the system 63

9 A Text Book of Operational Research and Food Plant Management L s = - = ρ 1 - ρ 5. Mean (expected or average) numbered of customer (units) in the queue waiting to service: L q = expected numbered in system - expected number in service station L q = L s - ρ = - - = 2 ( - ) = ρ ρ = = Ls = ρ (Ls) 6. Mean (expected) waiting time in system (time in queue plus service time) : Expected number in system - W s = Expected rate of arrival = = 1-7. Mean (expected or average) time a unit spends waiting in queue: Expected number in queue Expected rate of arrival 2 ( - ) = 1 - = Ws W q = = 8. The probability that the queue size in greater than or equal to k. P(n > k) = k 9. Mean (expected or average) length of non empty queue. A non empty queue refers that number of customer in the system should be at least two, one in queue and one under service station. The probability of non-empty queue is given by: P(n > 2) = Average length of non empty queue Averge length of a queue L nq = Probability of none empty queue 2 = ( - ) 2 = Average (expected) length of non empty system L ns = - + = Lnq + ρ 4.6 SOLVED EXAMPLES 1. A mechanist found that, it takes 1 2 hour to complete a ordered single place of work. Averagely he receives 10 orders per eight hour day. Calculate mechanics expected idle time each day and average number of jobs in the system assuming Poisson arrivals and exponential service times. Solution: Here, Mean arrival time () Mean service rate () = 10 orders per day = 10 orders/8 hours = 5/4 orders/hour = 1/2 hour per piece = 2 piece per hour Then, 1. Expected idle time of mechanist each day = Number of available hour - Hours by which mechanist is busy = 8-8 Traffic intensity (/) 5 4 = 8-8 = = 8-5 = 3 hours This can also be done as, 2 64

10 Queuing Theory (ii) Expected idle time of mechanist: = 8 hours probability that service facility is idle (P 0) = 8 (1 - /) = 8(1-5/8) = 3 hours Expected (or average) number of piece of work in system L s = - = 5/4 2-5/4 = 5 3 = 2 (approx) piece of work. 2. The data relating to dock shows that he average arrival rate of trucks is 2 per hour. The average time to load a truck using 3 loaders is 20 minutes, so that service rate is 3 per hour. With these data and assuming Poisson arrivals and exponential service rate calculate: (i) expected number of trucks in the system (ii) expected number of trucks waiting to be served (iii) expected time that a truck is in the system (iv) expected time in waiting line. (v) probability that, a truck has to wait few service (vi) probability that there is 5 units in system. Solution: Here = 2/hour and = 3/hour, then (i) The expected number of trucks in system (L s) = - = = 2 trucks (ii) The expected number of trucks waiting to be served (L q) 2 = ( - ) = Ls = = 4 3 trucks (iii) The expected time that a truck is in the system (Ws) 1 = - = = 1 hour (iv) The expected time in waiting line (W q) = Ws = = 2 3 hour (v) The probability that a truck has to wait for service (ρ) ρ = = 2 3 (vi) The probability that there are 5 units in system Have P 0 P 5 = P 0 n = P = Probability that that is no unit in the system, = 1 - / = 1-2/3 = 1 3 P 5 = = In a particular single server system, the arrival rate is 5 per hour, service rate is 8 per hour. Assume the condition for use of single channel queuing model, find out 65

11 A Text Book of Operational Research and Food Plant Management (a) (b) (c) the probability that the server is idle. the probability that there are at least 2 customers in the system. expected time that a customer is in the queue. Solution Here, = 5/hour, = 8/hour (a) The probability that the served in idle (ρ 0) P 0 = 1 - = = 3 8 (b) The probability that there are at least 2 customers in system is given by P(n > 2). P(n > 2) = 1 - (probability that there is no units in system plus probability that there is one unit is system) = = = = 0.39 (c) Expected time that a customer is in the queue (W q) W q = 1 - = = 5 24 hour 4. Customer arrives at the first class ticket counted of a film hall at a rate of 12 per hour. There is one clerk serving the customer at a rate of 30 per hour. Assuming the condition for the used single channel queuing model evaluate. a. Probability that there is no customer at the counted. b. Probability that there are more than 2 customers at the counter. c. Probability that there is no customer waiting to be served. d. Probability that a customer is being served and nobody is waiting. Solution: Here, = 12 / hour = 30 / hour a. Probability that there is no customer at the counted (P 0) P 0 = 1 - = = 0.6 b. Probability that there are more than 2 customers in the counted i.e. P (n > 3) P (n > 3) = 1 - (P 0 + P 1 + P 2) where P 0 = 1 -, P1 = (P0) and P 2 = (P 0) = 1 - [ (0.4) 2 ] = 1 - ( ) = c. Probability that there is no customer waiting to be served. This equal to probability that there is no customer in the system plus probability that there is one customer in system. P = P 0 + P 1 (P 0 and P 1 are determined on above question) or, P = ( ) = 0.84 d. Probability that a customer is being served and no body is waiting. This equal to probability that there is one customer in system i.e. P 1. 1 P 1 = (P 0) = = An university medical clinic has four full time physicians on hand to care for its large student population. A student who visits the clinic checks in at the reception desk and then waits until one of the four physicians is available. The

12 Queuing Theory service discipline is kept as first come - first served basis Assuming student arrival rate and a physician's service rate is 20 and 7 students per hour respectively and the condition for use of single channel queuing model calculate : (i) the average number of student in waiting room. (ii) the time an average student spends in the waiting room. (iii) the average number of student in clinic. (iv) the time an average student spends in the clinic. (v) the probability that there is no student in the clinic. (vi) the probability that there is 5 student in the clinic. (vii) the utilization factor of the clinic. Solution: Here, = mean arrival time = 20 student per hour = mean service time = 4 student per hour per doctor As there are 4 physicians, then = 7 4 = 28 student per hour. Then, (i) Average number of student in waiting room (L q) L q = 2 ( - ) = (20) 2 28 (28-20) = = students (ii) The time an average student spends in the waiting room W q = 1 - = = 5 56 hours = 5.35 minutes. = 5.36 minutes (iii) The average number of student in clinic (L s) L s = - = = 20 8 = students (iv) The time an average student spends in the clinic (W s) W s = 1 - = = 1 8 hours = 7.5 minutes (v) The probability that there is no student in clinic (P 0) P 0 = 1 - = = 8 78 = 0.29 (vi) The probability that there are 5 students in clinic (P 5) P 5 = P 0 5 = = (vi) The utilization factor of clinic (ρ) ρ = = = 0.71 (vii) ) ) the probability that a customer spends more than the average amount of time in clinic. Here, t and W s = 1 8 hours = hours : P(waiting time > t) or W s (t) = e -t/ws P(waiting time > hours) or W s (0.125 hours) = e /0.125 = e -1 = = A single channel queuing system has Poisson arrivals and exponential service time. The mean arrival rate is 6 per hour and the mean service rate is 10 per hour. Determine (i) the average time the customer will spend in the system (ii) the average length of the queue (iii) the utilization factor of system (iv) the expected proportion of time the facility remain idle (v) the average length of system 67

13 A Text Book of Operational Research and Food Plant Management Solution: = mean arrival rate = 6 per hour = mean seduce rate = 10 per hour, then (i) The average time, the customer will spend in the system (W s) 1 W s = - = = 1 4 = 0.25 hour (ii) The average length o the queue (L q) L q = 2 ( - ) = (6)2 10 (4) = = student (iii) The utilization factor of the system (ρ) ρ = = 6 10 = 0.6 (iv) The expected time the facility remain idle (P 0) P 0 = 1 - ρ = = 0.4 (v) The average length of system (L s) L s = - = 6 4 = customer 4.7 PROBLEMS 1. A radio mechanic on an average finds 5 customers coming his shop every hour for repairing their radio sets. He disposes of each of them with in 10 minutes on an average. The arrival and service time follow Poisson and exponential distribution respectively. Find (i) The proportion of time during which his shop remain empty. (ii) The average number of customer in his system. (iii) The average time spent by a customer in the queue and the service as well. (iv) The probability of finding at least one customer in his shop. (v) The average time spent in the queue by a customer. [Ans:(i) 0.17,(ii) 5 customers,(iii) 1 hour,(iv) 5/6 hour,(v) 0.83] 2. In a single channel queuing model with mean arrival rate and mean service rate of 20 and 50 per hour respectively, find (i) Average number of customer waiting in the system and queue (ii) Average time a customer spend in system and queue (iii) probability that there is no customer and 5 customer in system. [Ans: (i) 0.67, 0.267, (ii) 2 minute & 48 seconds (iii) 0.6, 0.004] 3. Workers come to a tool store room to enquire about the special tools for a particular job. The average time between the arrivals is 60 second and the arrivals are assumed to be Poisson distribution. The average service time is 40 seconds. Determine (i) Average queue length. (ii) Average length of non-empty queue. (iii) Average number of workers in the system (iv) Mean waiting time of an arrival in queue (v) Mean waiting time in system. Ans: (i) 1.33 workers (ii) 3 workers (ii) 2 workers (iv) 1.33 minutes (v) 2 minutes 4. Customers arrive at a box office window, being manned by a single individual according to Poisson input process with a mean of 30 per hour. The time required to serve a customer has an exponential distribution with a customer has an exponential distribution with a mean of 90 seconds. (a) Find the probability of their being no customer is the system. (b) Average numbered of customer in the system. (c) Fraction of time the service is busy i.e. traffic intensity. (d) Average waiting time of a customer who spends in system. 68

14 Queuing Theory 5 Hari Thapa found that the arrival rate of customers is 40 per hour and his service rate is one customer on every two minutes. Assuming poisson arrival rate and exponential service rate and single channel queuing model determine: (i) Average number of customer in queue. (ii) Average time a customer waits before he is served. (iii) Average time a customer spends on the service system. (iv) Utilization rate (V) Probability that no customers are in shop. Ans : (i) 0.5 (ii) 15 seconds (iii)1minute (iv) 0.33 (v) An automated car washing system has a poisson arrival rate of 4 cars per minutes and the machine has the constant washing speed of 6 cars per hour. Considering the constant service rate single channel model determine: (i) Average number of car in queue (ii) Average number of car in system (iii) Average time a car spent on queue Ans : (i) 0.67 (ii)1.33s (iii)0.25 minute 69

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