Waiting lines –Queuing Theory.

The study of waiting lines is one of the oldest and most widely used quantitative analysis techniques. Waiting lines are an everyday occurrence, affecting people shopping for groceries, buying petrol, making a bank deposit or simply waiting on the telephone for the first available airline telephone operator to answer. Queues, another term for waiting lines may also take the form of machines waiting to be repaired, trucks in line to be unloaded, or airplanes lined up on a runway for permission to take off.

The three basic components of a queuing process are: Arrivals, Service Facilities, and the Actual Waiting line.

So we are going to discuss how analytical models of waiting line can help managers evaluate the cost and effectiveness of service systems.

Characteristics of a Queuing System

We are going to take a look at the three parts of a Queuing System: (1) the Arrivals or inputs to the system (sometimes referred to as the Calling Population). (2) The Queue or the waiting line itself and (3) the Service facility. These three components have certain characteristics that must be examined before mathematical queuing models can be developed.

1- Arrival Characteristics

The input source that generates arrivals or customers for the service system has three major characteristics. It is important to consider the (a) Size of the calling population, (b) the pattern of arrivals (Statistical Distribution) at the queuing

system and (c) the behavior of the arrivals.

Size of the Population

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Population sizes are considered to be either Unlimited (Essentially Infinite) or limited (Finite). When the number of customers or arrivals on hand at any given moment is just a small portion of the potential arrivals, the calling population is considered unlimited or infinite. For practical purposes, examples of unlimited populations include cars arriving at car park, shoppers arriving at the supermarket, or students arriving to register for courses at a large university. When this is not the case modeling becomes very complex.

An example of a Finite population is a repair shop with only eight machines that might breakdown and require service.

Pattern of arrivals at the system

Customers either arrive at the service facility according to some known schedule (For example, one patient every 15 mins or one student for admission every half hour) or else they arrive randomly. Arrivals are considered random when they are independent of one another and their occurrence cannot be predicted exactly.

Frequently in queuing problems the number of arrivals per unit of time can be estimated by a probability distribution known as the Poisson Distribution. For any given arrival rate such as 2 customers per hour, or 4 trucks per minute, a discrete Poisson distribution can be established by using the formula

P(x) = e -

x

x! for x = 0, 1, 2, 3, 4, …

Where P(x) = probability of x arrivals

x = number of arrivals per unit of time

= average arrival rate

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e = 2.7183 (which is the base of the

natural logarithms)

With the help of the table at appendix c, these values are easy to compute. The figure illustrates the Poisson distribution for =2 and =4. This means that if the average arrival rate is =2 customers per hour, the probability of 0 customers arriving in any random hour is about 13 %, probability of 1 customer is about 27 %, 2 customers about 27 %, 3 customers about 18 %, 4 customers about 9% and so on. The chances that 9 or more will arrive are virtually nil. Arrivals are of course not always Poisson (they may follow some other distribution) and should be approximated by Poisson before the distribution is applied.

Behaviour of Arrivals

Most queuing models assume that an arriving customer is a patient customer. Patient customers are people or machines that wait in the queue until they are served and do not switch between lines.

Unfortunately life and quantitative analysis are complicated by the fact that people have been known to balk or renege.

Balking refers to customers who refuse to join the waiting line because it is too long to suit their needs or interest.

Renege customers are those who enter the queue but they become impatient and leave without completing their transaction.

How many times have you seen customers reaching the cash point with the caddies full of groceries including perishables abandoning the caddy before checking out because the waiting line was too long. Managers must be aware of these expensive occurrences to take the appropriate service level decisions.

2- Waiting Line Characteristics

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The Waiting line itself is the second component of the Queuing System. The length of a line can be either Limited or Unlimited. A queue is limited when it cannot by law of physical restrictions increase to infinite length. Example: a small restaurant that has only 10 tables and can serve no more than 50 people an evening. Analytical queuing models are treated under the assumption of unlimited queue length. A queue is unlimited when its size is unrestricted.

A second waiting line characteristic deals with Queue Discipline. This refers to the rule by which customers are serviced. Most systems use a queue discipline known as FIFO or FCFS. Example: in a hospital priority is given to emergencies. Patients who are critically injured move ahead in the treatment priority. Shoppers with less than 10 items may be allowed to enter an express checkout queue but are then treated as FCFS.

3- Service Facility Characteristics

The third part of any queuing system is the service facility. It is important to examine 2 basic properties: (1) the Design or configuration of the service systems and (2) the distribution of service times.

(1)Basic Queuing System Configurations or Designs

Service systems are usually classified in terms of their number of channels or number of servers and number of phases or number of service stops that must be made. A single-channel queuing system is a service system with one line and one server. An example is a small bank with just one cashier serving the customers. A multi-channel queuing system is a service system with one waiting line but with several servers. An example is a bank with several cashiers serving customers from one common waiting line. Most banks, airline ticket counters and post offices are multi-channel service systems.

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In a single-phase system, the customer receives services from only one station and then exits the system. A fast-food restaurant in which the person who takes your order also brings your food and takes your money is a single-phase system. However if the restaurant requires you to place your order at one station, pay at a second and pick up your food at a third station in this case it is a multi-phase system. Therefore the multi-phase system is a system in which the customer receives services from several stations before exiting the system. There are therefore 4 possible channel configurations.

1. A Single-channel, Single-phase system2. A Single-Channel, Multi-phase system3. A Multi-Channel, Single-phase system4. A Multi-Channel, Multi-phase system

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(2) Service Time DistributionService patterns are like arrival patterns in that they may be either constant or random. If service is constant, it takes the same amount of time to take care of each customer. This is the case in a machine-performed service operation such as an automatic car wash.More often service times are randomly distributed. In many cases it can be assumed that random service times are described by the negative exponential probability distribution.The figure illustrates that if service times follow an exponential distribution, the probability of any very long service time is low. For example, when an average service time is 20 minutes (or 3 customers per hour), seldom if ever, will a customer require more than 90 minutes or 1.5 hours in the service facility. If the mean service is 1 hour, the probability of spending more than 3 hours in service is virtually zero.

Queuing Costs

Operations Managers must recognize the trade-off that takes place between two costs: the cost of providing good service and the cost of customer or machine

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waiting time. Managers want queues that are short enough so that customers do not become unhappy and either leaves without buying or buy but never return. However managers may be willing to allow some waiting if it is balanced by the significant savings in service costs. One means of evaluating a service facility is to look at the total expected costs. Total cost is the sum of expected service costs plus expected waiting costs. See figure below:

From the figure, the service costs increase as a firm attempts to raise its level of service. This can be done by assigning more resources to the system to prevent or shorten excessively long lines. As the level of service improves (that is speeds up), however the cost of time spent waiting in lines decreases. The goal is to find the service level that minimizes the total expected cost.

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The variety of Queuing Models

A variety of Queuing Models may be applied in operations management. We will introduce 4 of the most widely used models. They all have 3 characteristics in common. They all assume:

1. Poisson distribution arrivals2. FIFO or FCFS discipline3. A single-service phase

In addition they all describe service systems that operate under steady, ongoing conditions. This means that arrival and service rates remain stable during the analysis.

Model A ( M/M/1): Single Channel Queuing Model with Poisson Arrivals and Exponential Service Times

The single-channel, single-phase model is one of the most widely used and simplest queuing model. It assumes that the following conditions exist in this type of system:

1. Arrivals are served on a FIFO basis, and every arrival waits to be served, regardless of the length of the line or queue.

2. The arrivals are independent of preceding arrivals, but the average number of arrivals (arrival rate) does not change over time.

3. Arrivals are described by Poisson probability distribution and come from an infinite population

4. Service times vary from one customer to the next and are independent of one another, but their average rate is known

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5. Service times occur according to the Negative Exponential Probability Distribution

6. The Service rate is larger than the arrival rate.

When these conditions are met, a series of equation can be developed that defined the queue’s operating characteristics. The mathematics used to derive these equations is very complex, so we will just use the derived equations.

Queuing Equations

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Example 1

A servicing shop is able to install a new car battery at an average rate of 3 per hour (or about 1 every 20 minutes), according to a negative exponential distribution. Customers arrive at the shop on the average of 2 per hour, following a Poisson distribution. They are served on a FCFS basis and come from an infinite (very large) population of possible buyers. Using this information we can obtain the operating characteristics of the shop queuing system:

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Once we have computed the operating characteristics of a queuing system, it is often important to do an economic analysis of their impact. Although, the model described above is valuable in predicting potential waiting times, queue lengths, idle times, and so on it does not identify optimal decisions or consider cost factors. Many times management have to make a trade-off between the increased cost of providing better service and the decreased waiting time costs derived from that service.

So let us work out the cost involved in example 1.

Suppose that it is estimated that the cost of customer waiting time, in terms of dissatisfaction and lost goodwill is $10 per hour of time spent waiting in line. Assume an 8 working hours day. What is the expected costs?

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Model B ( M/M/S): Multi-Channel Queuing Model with Poisson Arrivals and Exponential Service Times

Let us now look at a multiple channel queuing system in which two or more servers or channels are available to handle arriving customers. We will still assume that customers awaiting service form a single line and then proceed to the first available server. Multi-channel, single-phase waiting lines are found in many banks today: a common line is formed, and the customer at the head of the line proceeds to the first free server. This system assumes that arrivals follow a Poisson distribution and the service times are exponentially distributed. Service is FCFS, and all servers are assumed to perform at the same rate. All the assumptions listed before for the Single-channel also apply.

The equations used for this model is even more complex that the equations we used for the single-channel, but they are used in exactly the same fashion and provide the same type of information as the previous model.

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Example 2

The servicing shop has decided to hire a second mechanic to handle the installations. Customers, who arrive at the rate of 2 per hour, will wait in a single line until 1 of the 2 mechanics is free. Each mechanic installs batteries at the rate of about 3 per hour.

To find out how this system compares with the single channel waiting line system, we will compute several operating characteristics for the M=2 and compare the results with those found previously.

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Model C ( M/D/1) : Constant Service Time Model

Some service systems have constant, instead of exponentially distributed service times. When customers or equipment are processed according to a fixed cycle, as is the case of an automatic car wash, constant service times are appropriate. Because constant rates are certain, the values for Lq, Wq, Ls, and Ws are always less than they would be in Model A, which has variable service rates. As a matter of fact, both the average queue length and the average waiting time in the queue are halved with Model C. Constant service model formulas are given below:

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Example 3

A recycling company collects and compacts aluminium cans and glass bottles. Its truck drivers currently wait an average of 15 minutes before emptying their loads for recycling. The cost of driver and truck time while they are in queues is valued at $60 per hour. A new automated compactor can be purchased to process truckloads at a constant rate of 12 trucks per hour (that is 5 minutes per truck). Trucks arrive according to a Poisson distribution at an average rate of 8 per hour. If the new compactor is put in use, the cost will be amortised at a rate of $3 per truck unloaded. You have to conduct an analysis to evaluate the costs versus the benefits of the purchase.

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Model D : Limited Population Model

When there is a limited population of potential customers for a service facility, we need to consider a different queuing model. This model would be used, for example, if we were considering equipment repairs in a factory that has 5 machines, if we were in charge of maintenance for a fleet of 10 airplanes, or if we ran a hospital ward that has 20 beds. The limited population model allows any number of repair people (Servers) to be considered.

This model differs from the previous three queuing models because there is a dependent relationship between the length of the queue and the arrival rate. Let us illustrate an extreme situation: if a factory had 5 machines and all were broken down and awaiting repair, the arrival rate would drop to zero. In general, then, as the waiting time becomes longer in the limited population model, the arrival rate of customers or machine drops. The queuing formulas for the limited or finite population are given below. A different notation is used than what we have used for Models A, B and C. To simplify what can be a time-consuming exercise, finite

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queuing tables have been developed that determine D and F. D represents the probability that a machine needing repair will have to wait in line. F is a waiting- time efficiency factor. D and F are needed to compute most of the other finite model formulas.

Queuing formulas

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A small part of the published queuing tables is illustrated below.. The table provides data for a population of N=5

To use the table, we follow four steps:

1. Compute X (the service factor, where X= T/ (T+U).2. Find the value of X in the table and then find the line for M( where M is the

number of service channels).3. Note the corresponding values for D and F.4. Compute L, W, J, H, or whichever are needed to measure the service

system’s performance.

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Example 4

Past records indicate that each of 5 laser computer printers at the Mechanical Department needs repair after 20 hours of use. Breakdowns have been determined to be Poisson Distributed. The one technician on duty can service a printer in an average of 2 hours following an exponential distribution. Printer downtime costs $120 per hour. Technicians are paid $25 per hour. Should the department hire a second technician?

Assuming the second technician can repair a printer in an average of 2 hours, we can use the table (because there are N=5 machines in this limited population) to compare the costs of 1 versus 2 technicians.

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