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Operations Management Queuing Models - Lecture 6 (Chapter 8) Dr. Ursula G. Kraus 1/44 Review • National Cranberry 2/44 Agenda The Effect of Variability - Why do Queues form? - Performance Measures for Queuing Systems - Special Queuing Models 4/44 Goldratt's Production Game Station 1 4 Station 2 Station 3 4 4 Station 5 4 Source: Eliyahu M. Goldratt, 1992, “The Goal”, North River Press, 104-112. Station 4 4 5/44 Critical Assumptions from National Cranberry Truck arrivals are constant, evenly spaced over the 11 (12) hour period The mix between wet and dry berries is constant at 70/30 No variation in processing time at the individual processing stages 6/44 Post Office: Why Do Queues form? • Average inter-arrival time is 4 minutes • Average processing time is 3 minutes We are processing customers faster than they arrive so what's the problem? Source: Managing Business Process Flows (1999) 7/44 Post Office: Customer processing Customer # Average inter-arrival time is 4 minutes Average processing time is 3 minutes Time Time Source: Prof. K. Gue, Winter 03 8/44 Post Office: With constant processing time Customer # Average inter-arrival time is 4 minutes Average processing time is 3 minutes Time Time Source: Prof. K. Gue, Winter 03 9/44 Post Office: … and constant interarrival time Customer # Average inter-arrival time is 4 minutes Average processing time is 3 minutes Time Time Source: Prof. K. Gue, Winter 03 10/44 Reasons Why Queues form Call # Variability: – interarrival times – processing times – server availability 10 9 8 7 6 5 4 3 2 1 0 0 20 40 60 80 100 TIME (System) Utilization: = throughput/capacity Inventory (# of calls in system) 5 4 3 2 1 0 0 Source: Managing Business Process Flows (1999) 20 40 60 TIME 80 100 11/44 Telemarketing at L.L. Bean During some half hours, 80% of calls dialed received a busy signal. Customers getting through had to wait on average 10 minutes for an available agent. Extra telephone expense per day for waiting was $25,000. For calls abandoned because of long delays, L.L.Bean still paid for the queue time connect charges. L.L.Bean conservatively estimated that it lost $10 million of profit because of sub-optimal allocation of telemarketing resources. Source: Managing Business Process Flows (1999). Data based on 1988 sales. 12/44 Agenda The Effect of Variability - Why do Queues form? - Performance Measures for Queuing Systems - Special Queuing Models 13/44 Notation: Variability m = Mean V(X) = Variance s = Standard Deviation V ( X ) E ( X m )2 s V (X ) where X = Random Variable Source: Managing Business Process Flows (1999) 14/44 How can we measure variability? Should be a relative measure! Example: standard deviation s = 10 minutes low variability if mean (m) = 4 hours high variability if mean (m) = 5 minutes Coefficient of Variation: s C m Ratio of the standard deviation to the mean Source: Managing Business Process Flows (1999) 15/44 System Parameters Inputs of System – (Average) Interarrival times: Ti (Average) Interarrival rates: Ri (=1/Ti ) – (Average) Service times: Tp (Average) Service rates: Rp (=1/Tp ) System structure – Number of servers: c – Number of queues – Maximum queue length (buffer capacity K) Operating control policies – Queue discipline (FIFO), priorities Source: Managing Business Process Flows (1999) 16/44 Process Model for Queuing Systems: Call Center Incoming calls Ri Waiting Queue “buffer” size K R Queue Rb Blocked Calls Server Rp Answered Calls Ra Abandoned Calls Source: Managing Business Process Flows (1999) 17/44 Queuing System: Car Wash 18/44 Notation: Throughput Rate and Capacity (Average) Arrival rate: Ri = 1/Ti, where Ti is the average interarrival time (Average) Throughput rate: R = Ri - Rb - Ra = net arrival rate (Theoretical total) Processing capacity: Rp = c/Tp for c servers where Tp is the average processing time System Utilization: = R / Rp < 100% average fraction of time server is busy Source: Managing Business Process Flows (1999) 19/44 Notation: Inventory/Customers in System Average number of customers in the waiting line: Iq Average number of customers in process: Ip = c Average number of customers in the system (waiting and being served): I = Iq+ Ip Incoming calls I Order Queue “buffer” size K Ri R Queue Rb Blocked Calls Iq Server Answered Calls Rp Ip Ra Abandoned Calls Source: Managing Business Process Flows (1999) 20/44 Queuing System: Car Wash Iq Ip I 21/44 Little’s Law (I = R x T) T=I/R I = Iq + Ip waiting; in queue Source: Managing Business Process Flows (1999) in process (c) being served 22/44 Queue Length Formula - Approximation: 2(c+1) + Ci Cp ´ Iq 1 2 utilization effect 2 2 variability effect where Ci is the coefficient of variation for the interarrival times Cp is the coefficient of variation for the service times Source: Managing Business Process Flows (1999) 23/44 Agenda The Effect of Variability - Why do Queues form? - Performance Measures for Queuing Systems - Special Queuing Models - M/M/1 Queuing Model 24/44 Conditions: Poisson Distribution Patterns that occur most frequently in queuing models are Poisson and Exponential Distributions. The number of arrivals during a time period T yield a Poisson Distribution if the following conditions apply: Certain events are occurring at random over a continuous period of time (or interval of distance, region of area, etc). These events occur singly (one at a time), i.e. it is not possible for two events to occur exactly simultaneously. The events also occur independently of each other, i.e. the fact that an event has occurred (or not occurred) does not affect the chance of another event occurring. 25/44 Examples: Poisson Distributions (Processes) The number of web page requests arriving at a server The number of telephone calls arriving at a switchboard, or at an automatic phone-switching system The number of photons hitting a photodetector, when lit by a laser source The number of raindrops falling over a wide area The arrival of customers in a queueing system 26/44 Poisson Distribution of Arrivals and Services If arrivals at a service facility occur on a purely random fashion (one at a time), independently of each other then The number of arrivals during a time period T yield a Poisson Distribution The interarrival times yields an Exponential Distribution The Exponential and Poisson distributions can be derived from one another. The mean and standard deviation of the Exponential Distribution are equal: s m 27/44 M/M/1 Queue (Exponential Model) The M/M/1 is a single-server queue model, that can be used to approximate a lot of simple systems. It indicates a system where: 1. Arrivals are a Poisson Process with independently and exponentially distributed interarrival times 2. Service is a Poisson Process where service time is independently and exponentially distributed 3. There is one server Note: “M” stands for Markovian, a reference to the memoryless property of the exponential distribution 30/44 Queue Length for an M/M/1 Queue Iq 2(c +1) 1 C2+ C2 p i 2 •Independent, exponental Ti and Tp •Single server Iq Source: Managing Business Process Flows (1999) 2 1 32/44 M/M/1 Queue Average number of customers in the waiting line: 2 Iq 1 (exact result) Average number of customers in the system (waiting and being served) I Iq + Ip 2 1 + 1 Little’s Law: Average total time in the system (Flow Time) T I/R Source: Managing Business Process Flows (1999) Tq Iq / R Tp Ip / R 33/44 Utilization and Variability Drive Congestion Average Flow Time T T I/R Variability Increases Tp Utilization (ρ) Source: Managing Business Process Flows (1999) 100% 34/44 Example: Calling Center Consider a mail order company that has one customer service representative (CSR) taking calls. When the CSR is busy, the caller is put on hold. The calls are taken in the order received. Each minute a caller spends on hold costs the company $2 in telephone charges, customer dissatisfaction and loss of future business. In addition, the CSR is paid $20 an hour. Assume that calls arrive exponentially at the rate of one every 3 minutes. The CSR takes on average 2.5 minutes to complete the order. The time for service is also assumed to be exponentially distributed. (a) What is the capacity utilization? (b) How many customers will be on average on hold? (c) What is the average time spent on hold? (d) Estimate the average hourly cost of operating the call center. Source: Managing Business Process Flows (1999) 35/44 Expected costs Cost of Service Tradeoff Total cost (Service) Capacity Cost (Customer) Waiting Cost Level of service 37/44 Example: Ticket Outlet Customers arrive at a suburban ticket outlet at a rate of 14 per hour on Monday mornings. Selling the tickets and providing general information takes an average of 3 minutes per customer. There is one ticket agent on duty on Mondays. Assume exponential interarrival and service times. a) b) c) d) What percentage of time is the ticket agent busy? How many customers are in line, on average? How many minutes, on average, will a customer spend in the system? What is the average waiting time in queue? Source: Managing Business Process Flows (1999) 38/44 Performance Measures to Focus on Sales – Throughput (R) – Abandoning rate (Ra ) Cost – Capacity utilization () – Queue length (Iq) ; Total number in process (I) Customer Service – Waiting time in queue (Tq ); Total time in process (T) – Blocking rate (Rb ) Source: Managing Business Process Flows (1999) 41/44 Levers for System Improvements Increase process capacity Rp – adding more servers – working faster (decreasing Tp) Reduce waiting time Tq and queue length Iq – reduce variability – decrease arrival rate (not desirable in general) Source: Managing Business Process Flows (1999) 42/44 Example: Calling Center with 2 CSRs (2 servers) 2 phone numbers I. – Company hires a second CSR who is assigned a new telephone number. Customers are now free to call either of the two numbers. Once they are put on hold customers tend to stay on line since the other may be worse 50% Queue Server 50% Queue Server 1 phone number: pooling II. – Both CSRs share the same telephone number and the customers on hold are in a single queue Queue Servers Source: Managing Business Process Flows (1999) 43/44