Chapter 18 Waiting Lines McGraw-Hill/Irwin Copyright © 2009 by The McGraw-Hill Companies, Inc All Rights Reserved Chapter 18: Learning Objectives • You should be able to: – Explain why waiting lines form in systems that are underloaded – Identify the goal of queuing management – List the measures of system performance that are used in queuing – Discuss the assumptions of the basic queuing models presented – Solve typical problems 18-2 Queuing Theory • Queuing theory – Mathematical approach to the analysis of waiting lines – Applicable to many environments • • • • • • • Call centers Banks Post offices Restaurants Theme parks Telecommunications systems Traffic management 18-3 Simple Queuing System System Processing Order Calling population Arrivals Waiting line Service Exit 18-4 Queuing Models: Infinite Source • Four basic infinite source models – All assume a Poisson arrival rate Single server, exponential service time Single server, constant service time Multiple servers, exponential service time Multiple priority service, exponential service time 18-5 Infinite-Source Symbols Customer arrival rate Service rate per server Lq The average number of customers waiting for service Ls The average number of customer in the system r The average number of customers being served The system utilization Wq The average time customers wait in line Ws The average time customers spend in the system Service time P0 The probability of zero units in the system Pn The probability of n units in the system M The number of servers (channels) Lmax The maximum expected number waiting in line 18-6 Basic Relationships System Utilization M Average number of customers being served r 18-7 Basic Relationships • Little’s Law – For a stable system the average number of customers in line or in the system is equal to the average customers arrival rate multiplied by the average time in the line or system Ls Ws Lq Wq 18-8 Basic Relationships • The average number of customers – Waiting in line for service: – In the system: Lq Ls Lq r • The average time customers are – Waiting in line for service – In the system Wq Lq Ls Ws Wq 18-9 Single Server, Exponential Service Time • M/M/1 2 Lq P0 1 Pn P0 n Pn 1 n 18-10 Single Server, Constant Service Time • M/D/1 – If a system can reduce variability, it can shorten waiting lines noticeably – For, example, by making service time constant, the average number of customers waiting in line can be cut in half Lq 2 ( ) – Average time customers spend waiting in line is also cut by half – Similar improvements can be made by smoothing arrival rates (such as by use of appointments) 18-11 Multiple Servers (M/M/S) • Assumptions: – A Poisson arrival rate and exponential service time – Servers all work at the same average rate – Customers form a single waiting line (in order to maintain FCFS processing) 18-12 M/M/S M Lq P M 1! M n M M P0 n 0 n! M !1 M Ws M Wq PW Ws 1 18-13 Maximum Line Length • An issue that often arises in service system design is how much space should be allocated for waiting lines • The approximate line length, n, that will not be exceeded a specified percentage of the time can be determined using the following: log K ln K n or log ln where specified 1 percentage K Lq 1 18-14 Operations Strategy • Managers must carefully weigh the costs and benefits of service system capacity alternatives • Options for reducing wait times: – Work to increase processing rates, instead of increasing the number of servers – Use new processing equipment and/or methods – Reduce processing time variability through standardization – Shift demand 18-15 .. .Chapter 18: Learning Objectives • You should be able to: – Explain why waiting lines form in systems that are underloaded – Identify the goal of queuing management – List the... parks Telecommunications systems Traffic management 18- 3 Simple Queuing System System Processing Order Calling population Arrivals Waiting line Service Exit 18- 4 Queuing Models: Infinite Source •... waiting in line is also cut by half – Similar improvements can be made by smoothing arrival rates (such as by use of appointments) 18- 11 Multiple Servers (M/M/S) • Assumptions: – A Poisson arrival