Introduction to management science 10e by bernard taylor chapter 12

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