Bus Block II BA 301 Waiting Line Models

Bus. Block II BA 301 Waiting Line Models Quantitative Module D Power. Point presentation to accompany Operations Management, 6 E (Heizer & Render) 1 © 2001 by Prentice Hall, Inc. , Upper Saddle River, N. J. 07458

The CBA Advisement Center – Room 1 ¨ You need to get an override into a closed class. You are told to go to CBA Room 1 to see if you can get into the class. ¨ You arrive to find a line of 6 students waiting. As you wait, you and the person in front of you discuss the situation. ¨ It seems to both of you that there is always a line at the Advisement Center. 2 Power. Point presentation to accompany Operations Management, 6 E (Heizer & Render) © 2001 by Prentice Hall, Inc. , Upper Saddle River, N. J. 07458

CBA Advisement Center – Room 1 ¨ The person behind you says that she has been to the Advisement Center several times this week. On two occasions, there was no one in line and she got immediate service. There were lines of students on the other occasions and she had to wait for service. ¨ She thinks that the CBA should “do some research” to determine when students will show up for advisement services and have more people available to advise the students. ¨ She also complains that all she wants is to get her advisement folder and this takes less than a minute. When you tell her that you need to get an over-ride into a closed class, she predicts that it will take a lot longer to get that done. Power. Point presentation to accompany Operations Management, 6 E (Heizer & Render) 3 © 2001 by Prentice Hall, Inc. , Upper Saddle River, N. J. 07458

CBA Advisement Center – Room 1 ¨ Have you waited in line for service at the CBA Advisement Center? ¨ Have you ever walked in when there was no one ahead of you? ¨ Have you noticed that some student requests can be handled in a minute or less while other requests take “forever”? ¨ Does it seem like the students who take “forever” are just in front of 4 you? Power. Point presentation to accompany Operations Management, 6 E (Heizer & Render) © 2001 by Prentice Hall, Inc. , Upper Saddle River, N. J. 07458

Think about when you go to the bank or the grocery store, or call Dell Computer with a service question. ‘The other line always moves faster. ’ Thank you for holding. Hello. . . are you there? ‘If you change lines, the one you left will start to move faster than the one you’re in. ’ ‘You’re put on hold and listen all nine Beethoven symphonies while you’re waiting. Power. Point presentation to accompany Operations Management, 6 E (Heizer & Render) 5 © 1995 Corel Corp. © 2001 by Prentice Hall, Inc. , Upper Saddle River, N. J. 07458

Congratulations! ¨Welcome to the world of waiting lines! ¨Now - lets get some answers to our questions about waiting in lines. Power. Point presentation to accompany Operations Management, 6 E (Heizer & Render) 6 © 2001 by Prentice Hall, Inc. , Upper Saddle River, N. J. 07458

Waiting Lines ¨ First studied by A. K. Erlang in 1913 ¨ Analyzed telephone facilities ¨ Body of knowledge called queuing theory ¨ Queue is another name for waiting line ¨ Decision problem ¨ Balance cost of providing good service with cost of customers waiting Power. Point presentation to accompany Operations Management, 6 E (Heizer & Render) 7 © 2001 by Prentice Hall, Inc. , Upper Saddle River, N. J. 07458

Waiting Line Costs Cost cost line iting a al w ost Tot ce c i erv S Waiting time cost Level of service Optimal Power. Point presentation to accompany Operations Management, 6 E (Heizer & Render) 8 © 2001 by Prentice Hall, Inc. , Upper Saddle River, N. J. 07458

Waiting Line Examples Situation Arrivals Servers Bank Customers Teller Deposit etc. Process Doctor’s office Patient Doctor Service Treatment Traffic Cars Light Controlled intersection passage Assembly line Parts Workers CBA Advise- Students ment Center Power. Point presentation to accompany Operations Management, 6 E (Heizer & Render) Assembly Advisors Folders, Add. Drop, Overrides 9 © 2001 by Prentice Hall, Inc. , Upper Saddle River, N. J. 07458

Waiting Line Terminology ¨ Queue: Waiting line ¨ Arrival: 1 person, machine, part, etc. that arrives and demands service ¨ Queue discipline: Rules for determining the order that arrivals receive service ¨ Channel: Number of waiting lines ¨ Phase: Number of steps in service Power. Point presentation to accompany Operations Management, 6 E (Heizer & Render) 10 © 2001 by Prentice Hall, Inc. , Upper Saddle River, N. J. 07458

Figure D. 1 Power. Point presentation to accompany Operations Management, 6 E (Heizer &

Back to our CBA Advisement Center situation: ¨ What kind of information do we need to analyze the CBA Advisement Center’s front desk service? The pattern of the arrival of customers The behavior of customers How long it takes to serve a customer How the service system is designed The cost of providing the service The cost of a customer waiting in line ¨ We call of this the “characteristics of the waiting-line system”. Power. Point presentation to accompany Operations Management, 6 E (Heizer & Render) 12 © 2001 by Prentice Hall, Inc. , Upper Saddle River, N. J. 07458

CBA Advisement Center ¨ Suppose we know that on average, 90 students arrive each day (8 AM to 5 PM). ¨ Suppose we also know that it takes an average of 4 minutes to provide the service to a student. ¨ Is this enough information to analyze the waiting-line system? Power. Point presentation to accompany Operations Management, 6 E (Heizer & Render) 13 © 2001 by Prentice Hall, Inc. , Upper Saddle River, N. J. 07458

Characteristics of a Waiting Line System o pulati Po n ¨ Arrival rate distribution Servic e Facility Waiting Line Arrival Characteristics the source ¨ Size of population ¨ Poisson ¨ Other ¨ Pattern of arrivals ¨ limited ¨ unlimited ¨ Behavior of the arrivals ¨ join the queue, and wait ¨random ¨scheduled Power. Point presentation to accompany Operations Management, 6 E (Heizer & Render) 14 until served ¨ balk; refuse to join the line ¨ renege; by Prentice Hall, Inc. , Upper Saddle River, leave the line © 2001 N. J. 07458

Characteristics of a Waiting Line System continued Service Facility pulatio Po n Waiting Line Characteristics ¨Length of the queue ¨ limited ¨ unlimited ¨ Service priority ¨ FIFO ¨ other Power. Point presentation to accompany Operations Management, 6 E (Heizer & Render) 15 © 2001 by Prentice Hall, Inc. , Upper Saddle River, N. J. 07458

Characteristics of a Waiting Line System continued Service Facility pulatio Po Waiting Line n Service Facility Characteristics ¨ Number of ¨ Service time channels distribution ¨ single ¨ multiple ¨ negative exponential ¨ other ¨ Number of phases in service system ¨ single ¨ multiple Power. Point presentation to accompany Operations Management, 6 E (Heizer & Render) 16 © 2001 by Prentice Hall, Inc. , Upper Saddle River, N. J. 07458

CBA Advisement Center Waiting Line System Input source Service system Waiting line Service facility © 1995 Corel Corp. Power. Point presentation to accompany Operations Management, 6 E (Heizer & Render) 17 © 2001 by Prentice Hall, Inc. , Upper Saddle River, N. J. 07458

Poisson Distribution ¨ Number of events that occur in an interval of time = 0. 5 ¨ Example: Number of customers that arrive in 15 min. = 6. 0 ¨ Mean = (e. g. , 5/hr. ) ¨ Probability: Power. Point presentation to accompany Operations Management, 6 E (Heizer & Render) 18 © 2001 by Prentice Hall, Inc. , Upper Saddle River, N. J. 07458

Probabilit y Poisson Distributions for Arrival Times =2 Power. Point presentation to accompany Operations Management, 6 E (Heizer & Render) =4 19 © 2001 by Prentice Hall, Inc. , Upper Saddle River, N. J. 07458

Input Characteristics Input Source (Population) Arrival Pattern Size Infinite Finite Random Poisson Power. Point presentation to accompany Operations Management, 6 E (Heizer & Render) Behavior Non. Random Patient Impatient Other 20 © 2001 by Prentice Hall, Inc. , Upper Saddle River, N. J. 07458

Input Characteristics Input Source (Population) Arrival Pattern Size Infinite Finite Random Poisson Power. Point presentation to accompany Operations Management, 6 E (Heizer & Render) Behavior Non. Random Other 21 Patient Impatient Balk © 2001 by Prentice Hall, Inc. , Upper Saddle River, N. J. 07458

CBA Advisement Center “Balking” Input source Line was too long! Service system Service facility Waiting line © 1995 Corel Corp. Power. Point presentation to accompany Operations Management, 6 E (Heizer & Render) 22 © 2001 by Prentice Hall, Inc. , Upper Saddle River, N. J. 07458

Input Characteristics Input Source (Population) Arrival Pattern Size Infinite Finite Random Poisson Power. Point presentation to accompany Operations Management, 6 E (Heizer & Render) Behavior Non. Random Other 23 Patient Balk Impatient Renege © 2001 by Prentice Hall, Inc. , Upper Saddle River, N. J. 07458

CBA Advisement Center “Reneging” Input source Service system Service facility Waiting line I give up! © 1995 Corel Corp. Power. Point presentation to accompany Operations Management, 6 E (Heizer & Render) 24 © 2001 by Prentice Hall, Inc. , Upper Saddle River, N. J. 07458

Service Characteristics Service Facility Configuration Single Channel Multi. Channel Single Phase Power. Point presentation to accompany Operations Management, 6 E (Heizer & Render) 25 © 2001 by Prentice Hall, Inc. , Upper Saddle River, N. J. 07458

The CBA Advisement Center is a Single-Channel, Single-Phase System CBA Service system Arrivals Ships at sea Queue Service facility Ship unloading system Waiting ship line Served units Empty ships Dock Power. Point presentation to accompany Operations Management, 6 E (Heizer & Render) 26 © 2001 by Prentice Hall, Inc. , Upper Saddle River, N. J. 07458

Single-Channel, Multi-Phase System Service system Arrivals Cars in area Queue Service facility Mc. Donald’s drive-through Waiting cars Pay Power. Point presentation to accompany Operations Management, 6 E (Heizer & Render) 27 Served units Cars & food Pickup © 2001 by Prentice Hall, Inc. , Upper Saddle River, N. J. 07458

Multi-Channel, Single Phase System Service system Arrivals Service facility Queue Served units Service facility Example: Wells Fargo Bank customers wait in single line for one of several tellers. Power. Point presentation to accompany Operations Management, 6 E (Heizer & Render) 28 © 2001 by Prentice Hall, Inc. , Upper Saddle River, N. J. 07458

Multi-Channel, Multi-Phase System Service system Queue Service facility Arrivals Service facility Served units Service facility Example: At a laundromat, customers use one of several washers, then one of several dryers. Power. Point presentation to accompany Operations Management, 6 E (Heizer & Render) 29 © 2001 by Prentice Hall, Inc. , Upper Saddle River, N. J. 07458

Waiting-Line Performance Measures ¨ Average queue time, Wq ¨ Average queue length, Lq ¨ Average time in system, Ws ¨ Average number in system, Ls ¨ Probability of idle service facility, P 0 ¨ System utilization, ¨ Probability of k units in system, Pn > k Power. Point presentation to accompany Operations Management, 6 E (Heizer & Render) 30 © 2001 by Prentice Hall, Inc. , Upper Saddle River, N. J. 07458

Assumptions of the Basic Queuing Model ¨ Arrivals are served on a first come, first served basis ¨ Arrivals are independent of preceding arrivals ¨ Arrival rates are described by the Poisson probability distribution, and customers come from a very large population ¨ Service times vary from one customer to another, and are independent of one and other; the average service time is known ¨ Service times are described by the negative exponential probability distribution 31 ¨ The service rate is greater than the arrival Power. Point presentation to accompany Operations Management, 6 E (Heizer & Render) © 2001 by Prentice Hall, Inc. , Upper Saddle River, N. J. 07458

Types of Queuing Models ¨ Simple (M/M/1) ¨ Example: CBA Advisement Center ¨ Multi-channel (M/M/S) ¨ Example: Airline ticket counter ¨ Constant Service (M/D/1) ¨ Example: Automated car wash ¨ Limited Population ¨ Example: Department with only 7 drills Power. Point presentation to accompany Operations Management, 6 E (Heizer & Render) 32 © 2001 by Prentice Hall, Inc. , Upper Saddle River, N. J. 07458

Simple (M/M/1) Model Characteristics ¨ Type: Single-channel, single-phase system ¨ Input source: Infinite; no balks, no reneging ¨ Arrival distribution: Poisson ¨ Queue: Unlimited; single line ¨ Queue discipline: FIFO (FCFS) ¨ Service distribution: Negative exponential 33 Power. Point presentation to accompany Operations Management, 6 E (Heizer & Render) © 2001 by Prentice Hall, Inc. , Upper Saddle River, N. J. 07458

Simple (M/M/1) Model Equations Ls = Average number of units in queue - Ws = Average time in system 1 - 2 Lq = Average number of units in queue ( - ) = Average time in queue. Wq = System utilization Power. Point presentation to accompany Operations Management, 6 E (Heizer & Render) 34 © 2001 by Prentice Hall, Inc. , Upper Saddle River, N. J. 07458

Simple (M/M/1) Probability Equations Probability of 0 units in system, i. e. , system idle: P = 1 - = 10 Probability of more than k units in system: P = n>k () k+1 Where n is the number of units in the system Power. Point presentation to accompany Operations Management, 6 E (Heizer & Render) 35 © 2001 by Prentice Hall, Inc. , Upper Saddle River, N. J. 07458

Remember: & Are Rates ¨ = Mean number of If average service time is 4 minutes, then μ is 15 arrivals per time customers/hour period ¨ e. g. , 10 students/hour ¨ = Mean number of people or items served per time period ¨ e. g. , 15 Power. Point presentation to accompany Operations Management, 6 E (Heizer & Render) 36 © 1984 -1994 T/Maker Co. © 2001 by Prentice Hall, Inc. , Upper Saddle River, N. J. 07458

The Operating Characteristics of the CBA Advisement Center ¨ Based on the student arrival rate = 10 per hr. and the service rate =15 per hr. , calculate the following: Average time a student spends in the queue Average number of students in the queue Average time a student spends in the system Average number 37 students in the of Power. Point presentation to accompany Operations Management, 6 E (Heizer & Render) © 2001 by Prentice Hall, Inc. , Upper Saddle River, N. J. 07458

Cost Analysis of the CBA Student Advisement Center ¨ “Server” at the advisement desk is paid $40, 000 per year and works 2, 000 hours per year Cost per hour = $20 ¨ Cost of a student’s time while waiting in line = $6 per hour ¨ Total Cost of System = Sevice Cost + Waiting Cost TCS = (Number of Servers x Server Cost) + (Ave 38 Time Customer spends waiting in the system x Power. Point presentation to accompany Operations Management, 6 E (Heizer & Render) © 2001 by Prentice Hall, Inc. , Upper Saddle River, N. J. 07458