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

2 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.

3 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).

4 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.

5 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:, the mean arrival rate , the mean service rate , the number of servers

6 Information Flows   W = Average waiting time, per customer in the system W q = Average waiting time, per customer in the queue L = Average number of customers in the system L q = 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

7 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.

8 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.

9 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).

10 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/.

11 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.

12 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.

13 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., W q  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

14 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) = d. A city information service should be busy at least 60% of the day (i.e., P w > 0.6).

15 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.

16 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

17 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

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

19 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.

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