Systems Analysis Methods Dr. Jerrell T. Stracener, SAE Fellow

SMU EMIS 5300/7300 NTU SY-521-N Systems Analysis Methods Dr. Jerrell T. Stracener, SAE Fellow Queuing TheoryBasic Concepts and Models updated 11.07.01

Basic Concept • Service facilities are usually designed so that • their capacity is less than the maximum demand • Whenever demand exceeds capacity, a waiting • line, or a queue is formed; that is, the customers • do not get service immediately upon request but • must wait • On other occasions the service facility will be idle. • Management is interested in finding the • appropriate level of service

System Waiting for Serviceor Queue Servicing Serviced 1 2 x1 x2 x3 … xn ... K Arriving Customers Departing Customers Service Facility

Examples of Queuing Theory Application • Determining the capacity of an emergency room • in a hospital • Determining the number of runways at an • airport • Determining the number of elevators in a • building • Determining the number of traffic lights and • their frequency of operation • Determining the number of flights between • two cities

Examples of Queuing Theory Application • Determining the number of first-class seats • in an airplane • Determining the size of a restaurant • Determining the number of employees in a • storeroom, in a typing pool, or in a nursing team

The Structure of a Queuing System • The customers and their source - • Customers are defined as those in need of • service. Customers can be people, airplanes, • machines, or raw materials. The customers • are generated from a population or a source. • For example, a hospital’s ‘population’ would • be the sick requiring hospitalization. • The arrival process - The manner in which • customers show up at the service facility is • called the arrival process.

The Structure of a Queuing System • The service facility and the service process - • The service is provided by a service facility (or • facilities). This may be a person (a bank teller, • a barber), a machine (elevator, gasoline pump), • or a space (airport runway, parking lot, hospital • bed), to mention just a few. A service facility • may include one person or several people • operating as a team. • The queue - Whenever an arriving customer • finds that the service facility is busy, a queue, • or waiting line, is formed.

Some Examples of Queuing Systems System Queue Service Bank People Tellers Telephone Callers Switchboard Library Books to be shelved Librarian’s assistant Freeway Automobiles Tollbooth Airplane People Seats, flights Airport Circling planes Runways

Some Examples of Queuing Systems cost ($) 500 total cost 400 min 300 cost of providing service 200 cost of waiting 100 0 1 2 3 4 5 6 7 8 9 10 optimal level level of service

Costs Involved in a Queuing Situation • The facility cost - the cost of providing a service • includes: • 1. Cost of construction (capital investment) as • expressed by interest and amortization. • 2. Cost of operation: labor, energy, and materials • required for operations. • 3. Cost of maintenance and repair. • 4. Other costs: insurance, taxes, rental of space, • and other fixed costs.

Costs Involved in a Queuing Situation • The cost of waiting customers - the cost of • waiting time is more difficult to assess. It involves • several components. For example, a waiting • customer may get impatient and leave, thus • resulting in a loss of revenue and possible loss of • repeat business due to his or her dissatisfaction. • There may also be ‘ill-will’ cost incurred.

Management Objectives • Management may hold either or both of the • following objectives when making decisions • about an appropriate service level. • Cost minimization - In cases where it is • possible to ascribe a cost to the waiting time • (usually when a company is serving its own • employees or equipment), management will • provide a service level such that the total cost • of waiting and service is minimized.

Management Objectives • Achieving a specified performance level • (service goal) - Instead of (sometimes in addition • to) minimizing costs, management will strive to • achieve a certain level of service. For example: • - Telephone companies want to repair 99% of all • inoperative telephones within 24 hours. • - Fast-food restaurants advertise that you will • not have to wait more than three minutes. • - Banks try to avoid having more than six cars in • any lane of their drive in windows at a time. • - Service facilities should be in use at least 60% • of the time.

The Process • The managerial application of waiting line theory • involves the use of computed measures of • performance for selecting an alternative solution • to a queuing problem, usually among small • numbers of alternatives. The entire process • involves three steps: • Establish the measures of performance (or the • operating characteristics) of the queuing system. • Compute the measures of performance (result • variables) • Conduct an analysis

Establish the Measures of Performance • In this step a model of the problem is formulated • and the measures of performance are decided • upon. Examples of such measures are: • The average waiting time per customer • The average number of customers in the • waiting line • The utilization (busy period) of the service facility, or else its idle time.

Compute the Measures of Performance • Once the problem has been formulated, one of • two solution methods is employed to find the • measures of performance: • For problems in which certain theoretical • statistical distributions can describe the data, • formulas or equivalent tables can be used. • For other problems, Monte Carlo simulation is • used. • The measures of performance are then computed • for every course of action under consideration

The Analysis • In queuing analysis, there are usually only a • small number of alternatives to be evaluated. • For example, in a decision about the number of • elevators to be constructed in a new building, • 10 possibilities would be a realistic consideration, • but not 5000. The number of feasible alternatives • in a service system is usually small because of • human, technical, and legal constraints. • Alternatives may differ in the size of the facility, • the number of facilities, the speed of service, the • priorities given to customers or in the operating • procedures. For each alternative, the measures • of effectiveness must be computed.

The Analysis • The alternative solutions are then compared • on the basis of their overall effectiveness. One • approach here is the use of the total cost curve. • The major problem in this step may be cost • assessment. A queuing system usually involves • several measures of performance, and it is • necessary to establish a common denominator • (such as a total cost or a total utility) to • quantitatively compare the alternatives. In some • cases a qualitative comparison of the alternatives • is performed and no attempt is made to perform • a cost analysis.

The Analysis • In a limited number of cases the comparative • analysis leads to an optimal solution - for • example, a decision regarding the choice of the • proper number of identical service facilities. In • such cases, an explicit dollar value for the cost • of waiting must be specified. • Most frequently the analysis involves the • assessment of performance levels under different • system configurations. For example, if waiting time • per customer is important it is useful to know, for • each alternative configuration, how long customers • must wait.

Different Arrangement of Service Facilities a. single service facility b. multiple, parallel identical facilities (single queue) waiting line

Different Arrangement of Service Facilities c. multiple, parallel non-identical facilities express line regular lines

Different Arrangement of Service Facilities d. series of facilities

Different Arrangement of Service Facilities e. combination of facilities

Queue Discipline • A queue is formed whenever customers arrive and • the facility is busy. The characteristics of the • queue depend on rules and regulations that are • termed the queue discipline. The queue discipline • describes the policies that determine the manner • in which customers are selected for service. • Examples of some common disciplines are: • A priority system - Priority is given to selected • customers. For example, those with five items or • less in a supermarket can go to the express lane. • The handicapped and passengers with reservations • board airplanes first.

Queue Discipline • Emergency (preemptive priority) systems - • This is a system in which an important customer • not only has a priority in entrance but can even • interrupt a less important customer in the • middle of his or her service. For example, in an • emergency case in a hospital, the doctor may • leave the regular patient in the middle of the • treatment. That is, the regular patient is • preempted by the emergency one.

Queue Discipline • Last-in, first-served (LIFS) - Last arrivals are • served first. This system is commonly used with • parts and materials in a warehouse since it • reduces handling and transportation. • First-in, first-served (FIFS) - Customers that • arrive first are served first.

The Behavior in a Queue • Some interesting observations of human behavior • in queues are: • Balking - customers refusing to join a queue, • usually because of its length • Reneging - Customers tiring and leaving the • queue before they are served. • Jockeying - Customers switching between • waiting lines (a common scene in a supermarket)

The Behavior in a Queue • Combining, dividing - Combining or dividing • queues at certain lengths (e.g., in a supermarket • when a counter is closed or opened). • Cycling - Returning to the queue immediately • after obtaining service. (Children taking turns at • a playground or ore cars at a mine). • Note: In this text we assume that a customer • enters the system, stays in the line (if necessary), • receives service, and leaves. If a customer behaves • otherwise (according to any of the above • observations), the queuing system becomes very • complex, requiring simulation for analysis.

SMU EMIS 5300/7300 SMU EMIS 5300/7300 NTU SY-521-N NTU SY-521-N Systems Analysis Methods Dr. Jerrell T. Stracener, SAE Fellow Queuing Modeling and Analysis updated 11.16.01

Queuing Models • Several models exist, depending on the • structure of the system, the nature of arrivals, • the service policies, and the behavior of the • customers in the queue. • These queuing situations are commonly • designated X/Y/Z where X indicates the arrival • process, Y indicates the service process, and • Z the number of servers.

Queuing Models • Some queuing situations are: • 1. Single server (1), Poisson arrivals (M), • exponential service (M), called M/M/1. • 2. Single server, Poisson arrival, exponential • service, with finite (limited) queue length: • M/M/1 finite queue. • 3. M/M/1 finite source (a finite calling population). • 4. M/arbitrary/1 (arbitrary service time distribution, • but mean and standard deviation are known). • 5. M/M/K (multiple servers: K). • 6. M/M/K finite queue. • 7. M/M/K finite source. • 8. M/constant/K (constant service times).

Solution Approaches • There are two basic approaches to the solution • of queuing problems: analytical and simulation. • The analytic approach - The measures of • performance are determined through the use of • formulas. Unfortunately, many queuing situations • are so complex that the analytic approach is • completely impractical or even impossible. • Simulation - For those situations in which the • analytic approach is unsuitable, the procedure of • simulation can be used.

Information Flow in Waiting Line Models • It is helpful to use some measures of performance • when evaluating service alternatives, particularly • when a cost approach is planned. • A solution to a queuing problem means computing • certain measures of performance • These measures are computed from three input • variables: • l, the mean arrival rate • m, the mean service rate • , the number of servers

Information Flows W = Average waiting time, per customer in the system Wq = Average waiting time, per customer in the queue L = Average number of customers in the system Lq = Average number of customers in the queue P(0) = Probability of the system being idle Pw = Probability of the system being busy P(t > T) = Probability of waiting longer than time T P(n) = Probability of having exactly n customers in the system P(n > N), P(n < N) - Probability of finding more than, or less than, N customers in the system m l 

Deterministic Queuing Systems The simplest and the rarest of all waiting line situations involves constant arrival rates and constant service times. Three cases can be distinguished: 1. Arrival rate equals service rate. Assume that people arrive every 10 minutes, to a single server, where the service takes exactly 10 minutes. Then the server will be utilized continuously (100% utilization), and there will be no waiting line.

Deterministic Queuing Systems 2. Arrival rate larger than service rate. Assume that there are six arrivals per hour (one every 10 minutes) and the service rate is only five per hour (12 minutes each). Therefore, one arrival cannot be served each hour, and a waiting line will build up (at a rate of one per hour). Such a waiting line will grow and grow as time passes and is termed explosive.

Deterministic Queuing Systems 3. Arrival rate smaller than service rate. Assume that there are again six arrivals per hour but the service capacity is eight per hour. In this case the facility will be utilized only 6/8 = 75% of the time. There will never be a waiting line (if the arrivals come at equal intervals).

The Basic Poisson-Exponential Model (M/M/1) The classical and probably best known of all waiting line models is the Poisson-exponential single server model. It exhibits the following characteristics. Arrival rate - The arrival rate is assumed to be random and is described by Poisson distribution. The average arrival rate is designated by the Greek letter . Service time - The service time is assumed to follow the negative exponential distribution. The average service rate is designated by the Greek letter , and the average service time by 1/.

The Major Ground Rules for the Operation of a • Single Server System • Infinite source of population • First-come, first-served treatment • The ratio / is smaller than 1. This ratio is • designated by the Greek letter . The ratio is a • measure of the utilization of the system. If the • utilization factor is equal to or larger than 1, the • waiting line will increase without bound (will be • explosive), a situation which is unacceptable to • management.

The Major Ground Rules for the Operation of a • Single Server System • Steady state (equilibrium) exists. A system is • in a ‘transient state’ when its measures of • performance are still dependent on the initial • conditions. However, our interest is in the ‘long • run’ behavior of the system, commonly known as • steady state. A steady state condition occurs when • the system becomes independent of time. • Unlimited queuing space exists.

Managerial Use of the Measures of Performance Some of these measures can be used in a cost analysis, while others are used to aid in determining service level policies. For example: a. A fast-food restaurant wants to design its service facility such that a customer will not wait, on the average, more than two minutes (i.e., Wq 2 minutes) before being served. b. A telephone company desires that the probability of any customer being without telephone service more than two days be 3% (i.e., P(t > 2 days) = 0.03

Managerial Use of the Measures of Performance c. A bank’s policy is that the number of customers at its drive-in facility will exceed 10 only 5% of the time (i.e., P(n > 10) = 0.05. d. A city information service should be busy at least 60% of the day (i.e., Pw > 0.6).

Example - The Toolroom problem The J.C. Nickel Company toolroom is staffed by one clerk who can serve 12 production employees, on the average, each hour. The production employees arrive at the toolroom every six minutes, on the average. Find the measures of performance.

Example - The Toolroom problem solution It is necessary first to change the time dimensions of  and  to a common denominator.  is not given in minutes,  in hours. We will use hours as the common denominator. 1. Average waiting time in the system (toolroom) hours, per employee 2. The average waiting time in the line. hours, per employee

Example - The Toolroom problem solution 3. The average number of employees in the toolroom area employees 4. The average number of employees in the line. employees

Example - The Toolroom problem solution 5. The probability that the toolroom clerk will be idle. 6. The probability of finding the system busy.

Example - The Toolroom problem solution 7. The chance of waiting longer than 1/2 hour in the system. That is T = 1/2. 8. The probability of finding four employees in the system, n = 4.

Example - The Toolroom problem solution 9. The probability of finding more than three employees in the system.

SMU EMIS 5300/7300 SMU EMIS 5300/7300 NTU SY-521-N NTU SY-521-N Systems Analysis Methods Dr. Jerrell T. Stracener, SAE Fellow Queuing Models and Equations updated 11.20.01

Example The Comtec Corporation is considering one of two possible self-service duplicating machines. The Mark I is capable of duplicating, on the average, 20 jobs each hour at a cost of $50 per day. Alternatively, the Mark II can duplicate, on the average, 24 jobs per hour at a cost of $80 per day. The duplicating center is open 10 hours a day with an average arrival of 18 jobs per hour. The duplication is performed by employees arriving from various departments, whose average hourly wage is $5. Should the company lease Mark I or Mark II?

Systems Analysis Methods Dr. Jerrell T. Stracener, SAE Fellow