Queuing Analysis Chapter 13 Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall 13-1 Chapter Topics ■ Elements of Waiting Line Analysis ■ The Single-Server Waiting Line System ■ Undefined and Constant Service Times ■ Finite Queue Length ■ Finite Calling Problem ■ The Multiple-Server Waiting Line ■ Additional Types of Queuing Systems Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall 13-2 Overview Significant amount of time spent in waiting lines by people, products, etc Providing quick service is an important aspect of quality customer service The basis of waiting line analysis is the trade-off between the cost of improving service and the costs associated with making customers wait Queuing analysis is a probabilistic form of analysis The results are referred to as operating characteristics Results are used by managers of queuing operations to make decisions Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall 13-3 Elements of Waiting Line Analysis (1 of 2) Waiting lines form because people or things arrive at a service faster than they can be served Most operations have sufficient server capacity to handle customers in the long run Customers however, not arrive at a constant rate nor are they served in an equal amount of time Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall 13-4 Elements of Waiting Line Analysis (2 of 2) Waiting lines are continually increasing and decreasing in length and approach an average rate of customer arrivals and an average service time, in the long run Decisions concerning the management of waiting lines are based on these averages for customer arrivals and service times They are used in formulas to compute operating characteristics of the system Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall 13-5 The Single-Server Waiting Line System (1 of 2) Components of a waiting line system include arrivals (customers), servers, (cash register/operator), customers in line form a waiting line Factors to consider in analysis: The queue discipline The nature of the calling population The arrival rate The service rate Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall 13-6 The Single-Server Waiting Line System (2 of 2) Figure 13.1 Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall 13-7 Single-Server Waiting Line System Component Definitions Queue Discipline: The order in which waiting customers are served Calling Population: The source of customers (infinite or finite) Arrival Rate: The frequency at which customers arrive at a waiting line according to a probability distribution (frequently described by a Poisson distribution) Service Rate: The average number of customers that can be served during a time period (often described by the negative exponential distribution) Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall 13-8 Single-Server Waiting Line System Single-Server Model Assumptions of the basic single-server model: An infinite calling population A first-come, first-served queue discipline Poisson arrival rate Exponential service times Symbols: λ = the arrival rate (average number of arrivals/time period) µ = the service rate (average number served/time period) Customers must be served faster than they arrive (λ Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall 13-9 Single-Server Waiting Line System Basic Single-Server Queuing Formulas (1 of 2) Probability that no customers are in the queuing system: ữ P0 = ữ Probability that n customers are in the system: n n ữ ìP = λ ÷ 1− λ ÷ Pn = àữ àữ àữ L= λ µ −λ Average number of customers in system: λ Lq = µ µ − λ ÷ Average number of customer in the waiting line:13- Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Multiple-Server Waiting Line Queuing Formulas (2 of 3) P0 = n c n=c−1 λ ÷ c ữ n! µ ÷ + c! µ ÷ cµ − λ ÷÷÷ n=0 = probability no customers in system n Pn = nc ữữ P0 for n > c c!c n Pn = n ữữ P0 for n < c = probability of n customers in system c λµ ( λ / µ ) L= P0 + λ µ = average customers in the system µ (c −1)!(cµ − λ ) W = L = average time customer spends in the system λ Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall 13- Multiple-Server Waiting Line Queuing Formulas (3 of 3) Lq = L − λ µ = average number of customers in the queue Lq Wq =W − µ = = average time customer is in the queue λ c Pw = ữữ cà P0 = probability customer must wait for service c! c µ − λ Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall 13- Multiple-Server Waiting Line Biggs Department Store Example (1 ofλ = 2)10, µ = 4, c = P0 = 10 ÷ + 10 ÷ + 10 ÷ + 10 ÷ 3(4) 0! ÷ 1! ÷ 2! ÷ 3! ÷ 3(4) −10 = 045 probability of no customers (10)(4)(10/4) L= (.045) + 10 (3 −1)![3(4) −10]2 = customers on average in service department W = = 0.60 hour average customer time in the service department 10 Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall 13- Multiple-Server Waiting Line Biggs Department Store Example (2 of 2) Lq = − 10 = 3.5 customers on the average waiting to be served Wq = 3.5 10 = 0.35 hour average waiting time in line per customer 3(4) (.045) Pw = 10 3! 3(4) −10 ÷ ÷ = 703 probability customer must wait for service Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall 13- Multiple-Server Waiting Line Solution with Excel Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Exhibit 13.11 13- Multiple-Server Waiting Line Solution with Excel QM Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Exhibit 13.12 13- Multiple-Server Waiting Line Solution with QM for Windows Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Exhibit 13.13 13- Additional Types of Queuing Systems (1 of 2) Figure 13.4 Single Queues with Single and Multiple Servers in Sequence Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall 13- Additional Types of Queuing Systems (2 of 2) Other items contributing to queuing systems: Systems in which customers balk from entering system, or leave the line (renege) Servers who provide service in other than first-come, first-served manner Service times that are not exponentially distributed or are undefined or constant Arrival rates that are not Poisson distributed Jockeying (i.e., moving between queues) Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall 13- Example Problem Solution (1 of 5) Problem Statement: Citizens Northern Savings Bank loan officer customer interviews Customer arrival rate of four per hour, Poisson distributed; officer interview service time of 12 minutes per customer Determine operating characteristics for this system Additional officer creating a multipleserver queuing system with two channels Determine operating characteristics for this system Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall 13- Example Problem Solution (2 of 5) Solution: Step 1: Determine Operating Characteristics for the Single-Server System λ = customers per hour arrive, µ = customers per hour are served Po = (1 - λ / µ) = ( – / 5) = 20 probability of no customers in the system L = λ / (µ - λ) = / (5 - 4) = customers on average in the queuing system Lq = λ / µ(µ - λ) = 42 / 5(5 - 4) = 3.2 customers on average in the waiting Copyright © 2010 Pearsonline Education, Inc Publishing as Prentice Hall 13- Example Problem Solution (3 of 5) Step (continued): W = / (µ - λ) = / (5 - 4) = hour on average in the system Wq = λ / µ(u - λ) = / 5(5 - 4) = 0.80 hour (48 minutes) average time in the waiting line Pw = λ / µ = / = 80 probability the new accounts officer is busy and a customer must wait Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall 13- Example Problem Solution (4 of 5) Step 2: Determine the Operating Characteristics for the Multiple-Server System λ = customers per hour arrive; µ = customers per hour served; c = P0 = servers n c cµ n = c −1 λ λ ÷ ÷ + ÷ ữ ữ c! µ cµ − λ ÷÷ n ! n= = 429 probability no customers in system c λµ ( λ / µ ) L= P0 + λ µ µ (c −1)!(cµ − λ )2 = 0.952 average number of customers in the system Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall 13- Example Problem Solution (5 of 5) Step (continued): Lq = L − λ µ = 0.152 average number of customers in the queue = Lq Wq =W − µ λ = 0.038 hour average time customer is in the queue c cµ Pw = λ Po µ c! cµ − λ ÷ ÷ = 229 probability customer must wait for service Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall 13- Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall 13- ... Components of a waiting line system include arrivals (customers), servers, (cash register/operator), customers in line form a waiting line Factors to consider in analysis: The queue discipline ... customer in the queuing system Lq = 0.27 customer in the waiting line W = 0.055 hour per customer in the system Wq = 0.022 hour per customer in the waiting line U = 40 probability that a customer... waiting customers are served Calling Population: The source of customers (infinite or finite) Arrival Rate: The frequency at which customers arrive at a waiting line according to a probability