production and operations management- Waiting Line Management notes PDF

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Waiting Line Management A waiting line is one or more customers waiting for service. The customers can be people or objects, such as machines requiring maintenance, sales orders waiting for shipping or inventory items waiting to be used. A waiting line forms because of a temporary imbalance between the demand for service and the capacity of the system to provide the service. In most real life waiting line problems, the demand rate varies; that is the customers arrive at unpredictable intervals. Most often the rate of producing the service also varies, depending on the customer needs. Suppose the bank customers arrive at an average rate of 20 customers per hour throughout the day and that the bank can process an average of 20 customers per hour, then also a waiting line may be formed. Because the customer arrival rate varies throughout the day and the time required to process a customer may vary. During the noon hour, 30 customers may arrive at the bank. Some of them may have complicated transactions, requiring above average process times. Waiting line can develop even if the time to process a customer is constant. For example, a subway train is computer controlled to arrive at stations along its route. Each train is programmed to arrive at a station, say every 15 minutes. Even with the constant service time, waiting line develops while riders wait for the next train or cannot get on a train because of the size of the crowd at a busy time of the day. Economics of Waiting Line In a Fast Food Restaurant during peak meal times each day, there is a temporary surge in demand that can be quickly handled with the available capacity. A fast food restaurant experiences both variable demand and service time. 1

In a waiting line system, some type of compromise between queue size, the waiting time and the cost of the service, have to be made. Organizations design their waiting line system considering the consequences of having a customer wait in line (which leads to customer dissatisfaction and lost future business), versus the cost of providing more service capacity. This is an important tradeoff and therefore management must consider: what is the optimum level of service that they should provide.

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Initially with the minimal service capacity, the waiting line cost is minimum. As service capacity is increased there is a reduction in the number of customers in line and their waiting times, which decreases waiting line costs. The Waiting Line or Queuing system consists of three major components. 1. The source population and the way customer arrive at the system. 2. The servicing or queuing system. 3. The condition of the customers exiting the system (back to source population or not).

1. Customer Arrival Arrival at a service system may be drawn from a finite or infinite population. Finite Population: it refers to the limited size customer pool that will use the service and at times, form a line. In a finite population when a customer leaves its 3

position as a member for the population (a machine breakdown and requiring service, for example), the size of user group is reduced by one, which reduces the probability of next occurrence. Conversely, when a customer is serviced and returns to the user group, the population increases and the probability of user requiring service also increases. As an example, consider a group of six machines maintained by one repair person. When one machine breaks down, the source population is reduced to five, and the chance 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 machine is repaired and returned from service, the machine population increases, thus raising the probability of next break down. Infinite Population: it is large enough 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 retaining to the population) does not affect the system probabilities. If in the preceding finite population there were 100 machines instead of six, then if two or more machine break down, the probabilities for the next break down would not be very different. Distribution of Arrivals Arrival Rate: the number of units per period (e.g. as an average of one every six minute) In observing arrival at a service facility, we can look at them from two view points: first we can analyse the time between successive arrivals, in this case we assume 4

that the time between arrivals is exponentially distributed. In the second case, we can get some time length (T) and try to determine how many arrivals might enter the system within T. In this case we assume that number of arrivals per time period is Poisson distributed. Arrival Patterns: Arrival Pattern may be controllable or uncontrollable. Controllable (for example A Barber may reduce Saturday’s arrival by charging extra for the same service, a departmental store may run sales during off seasons etc.) Some service demands are uncontrollable, such as emergency medical demands. But even in these situations, arrivals at emergency rooms in specific hospitals are controllable to some extent by keeping ambulance drivers in the service region informed of the status of their respective host hospitals. Degree of Patience: A patient arrival is one who waits as long as necessary until the service facility is ready to serve him. There are two classes of impatient arrivals. Members of the first class survey both the service facility and the length of the line and decide not to enter the waiting line. The behavior of this type is known as Balking. Example is when we go to the telephone bill payment counter and see that there are already 30 persons waiting to pay their bills. So we choose to come back later. Those in the second class arrive, enters the waiting line but leaves before being serviced. For example holding the telephone lines long enough, we decide to leave and thereby disconnect the line. This is known as Reneging. 2. The Queuing System: factors 5

Length: A waiting line may have infinite potential length or may have limited capacity length. An infinite line is one that is very long in terms of capacity of the service system. For example line of vehicles backed up for miles at a bridge crossing and customers waiting to purchase tickets at a theatre. Gas stations, Loading docks and parking lots have limited capacity caused by legal restrictions or physical space characteristics. Number of Lines: may be single or multiple. The term multiple means the single lines that form in front of two or more servers. Queue Discipline: it is priority rule for determining the order of service to the customers in waiting line. They may be FCFS, LCFS, and Truncated Queue (in this queuing situation there is a limited waiting area. When this maximum queue length is reached, all further arrivals leave), Reservation First, Emergency First, Highest profit customer first, largest order first, longest waiting time in line and soonest promised date etc. Service time distribution Service rate: it is the capacity of the server in number of units per time period. (Such as 12 completions per hour). Service time can be constant or exponential. A constant service time means that each service takes exactly the same time. This characteristic is limited to machine controlled operations. When service time is random, it can be approximated by the exponential distribution. When the service time is exponentially distributed, we will refer to µ as the average number of units or customers that can be served per time period.. Line Structure 6

Single Channel (server), Single Phase: This is the simplest structure for example the cashier at a bank counter, any production line with a unidirectional material flow. Example is one person barber shop, single booth, theater ticket sales etc. Single Channel Multiphase: In this structure, the service is completed in a series of steps. A car wash is an illustration because a series of services vacuuming, weting, washing, rinsing, drying, window cleaning and packing is performed in a fairly sequence. A critical case in this system is that the amount of buildup of items allowed in front of each service which in turn constitute the separate waiting line.

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Multi Channel (Server) Single Phase: Teller’s window in a bank and billing counter in high volume departmental stores are the examples of this type of structure. The difficulty with this structure is that the uneven service time given to each customers being served before others who arrived earlier as well as in some degree of line shifting. Varying this structure to ensure the servicing of arrivals in chronological order would require forming a single line from which, as a server becomes available, the next customer in the queue is assigned. For example Railway Reservation counters. The major problem of the structure is that it requires rigid control of the lines to maintain order and to direct customers to available servers.

Multi channel Multiphase: this case is similar to the preceding one except that two or more services are performed in sequence of steps. For Example the admission of patients in a hospital follows this pattern because a specific sequence is usually followed: initial contact at the admission desk, filling out 8

forms, making identification tags, obtaining a room assignment, escorting the patient to the room and so forth. Because several servers are usually available for this procedure more than one patient at a time may be processed.

3. Exit: once a customer is served two exits are possiblea. The customer may return to the source population and immediately become a competing candidate for service again. b. There may be low probability of re service. The first case can be illustrated by a machine that has been routinely repaired and returned to duty but may break down again. The second case is illustrated by a machine that has been overhauled or modified and has a low probability of re service over the near future.

Line Length: the total number of customers in the system who are actually waiting in the line and not being served. Notations n = number of customers in the system (waiting and in service) Pn = probability of n customers in the system Λ =mean customer arrival rate or average number of arrivals in the queuing system per unit of time.

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µ = mean service rate or average number of customers completing service per unit of time. λ/µ = traffic intensity or server utilization factor S = number of service facilities/channels N=maximum number of customers allowed in the system (waiting + in service) Ls = mean number of customers in the system (waiting + in service) Lq = mean number of customers in the queue (queue length) Ws = mean waiting time in the system (waiting + in service) Wq = mean waiting time in the queue Pw = probability that an arriving customer has to wait.

Relationship Expected no. of customers in the system = expected no. of customers in queue + in service

Ls = Lq + λ/µ

Ls = λ Ws and Lq = λ Wq Classification of Queuing models: Standard format of queuing model to describe the characteristics of parallel queue. {(a/b/c) : (d/e)} Where a = arrivals distribution 10

b = service time distribution c = no. of service channels d = maximum number of customers allowed in the system (in queue plus in service) e = queue discipline M = exponential service time distribution inter arrival times or equivalently poisons’ arrival or departure distribution) Model 1 {(M / M / 1): (∞ / FCFS)} Single server, unlimited queue model – It is based on certain assumptions: 1. Exponential distribution of inter arrival times or poisson distribution of arrival rate. 2. Single waiting line with no restriction on length of queue (infinite capacity) and no balking and reneging. 3. Queue discipline is FCFS. 4. Single server with exponential distribution of service times. Formulas: Probability of no customer in the system P0 = 1- λ / µ Probability of n customers in the system = ( λ / µ ) n (1- λ / µ) Probability of server being busy = λ / µ Expected number of customers in the system L s = λ / (µ - λ)

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Expected number of customers waiting in the queue L q = λ2 / µ (µ - λ) Expected waiting time of a customer in the queue Wq = Lq / λ = λ / µ (µ - λ) Expected waiting time of a customer in the system Ws = Ls / λ = 1/ (µ - λ)

Examples: single booth theater ticket sales, Single teller bank window. Model 2 {(M / D / 1): (∞ / FCFS)} Single server queue with constant service time, unlimited queue model – It is based on certain assumptions: 1. Exponential distribution of inter arrival times or poisson distribution of arrival rate. 2. Single waiting line with no restriction on length of queue (infinite capacity). 3. Queue discipline is FCFS. 4. Single server queue with constant service times. Formula:

Examples: machine controlled manufacturing operation, automatic car wash system, roller coaster rides in amusement park, subway train. Model 3 {(M / M / 1): (N / FCFS)} Single server queue with exponential service time, limited queue model –

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Assumptions: It is based on certain assumptions: 1. Exponential distribution of inter arrival times or poisson distribution of arrival rate. 2. Single waiting line with limited length of queue (finite capacity). 3. Queue discipline is FCFS. 4. Single server queue with exponential service times. Examples: parking lots of retail store, manufacturing operation with in-process inventory. Model 4 {(M / M / S): (∞ / FCFS)} Multiple server queue with exponential service time and unlimited queue model – It is based on certain assumptions: 1. Exponential distribution of inter arrival times or poisson distribution of arrival rate. 2. Single waiting line with no restriction on length of queue. 3. Queue discipline is FCFS. 4. Multiple servers queue with exponential service times. Examples: customer care service, toll road pay booths, bank teller windows.

Q. How to manage queues which go beyond the waiting lines models?

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Questions of waiting line management

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1. Indian Overseas Bank is considering opening a cash transaction window for customer service. Management estimates that customer will arrive at the rate of 15 per hour. The teller who will staff the window can service customers at the rate of every three minutes. Assuming Poisson’s arrival and exponential service, find:  Utilization of teller  Average number of customers in the waiting line  Average number of customers in the system  The average waiting time in the line  The average waiting time in the system 2. The teller facility of State Bank of India at a college extensive counter has a one man operation at present. Customers arrive at the bank at the rate of every six minutes to use the teller. The service time varies randomly across customer account of some parameters. However based on the past, it has been found that the teller takes on an average 5 min to serve the customers. The arrival follow Poisson’s distribution and service time follow exponential distribution, find:  The average number of customers in the waiting line  The average number of customers in the system  Probability of having three customers in the system 3. Customer arrive at a sales counter managed by a single person according to a Poisson process with a mean rate of 20/hr. the time required to serve a customer has an exponential distribution with a mean of 30 customers/hr. Find the average waiting time of customer in queue, average waiting time of customer in the system and average service time. 15

4. The arrival of vehicles in a petrol pump station can be assumed to be follow Poisson distribution. The number of vehicles arriving in the station in an hour is 125. The station can attend around 150 vehicles on an average per hour. Find Probability that a customer has to wait for service Mean length of the system Mean length of queue Mean waiting time in system

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