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Presentation transcript:

Operations Management Queuing (Part 2) - Lecture 7 (Chapter 8) Dr. Ursula G. Kraus

Review Performance Measures for Queuing Systems Special Queuing Model: M/M/1

Learning Objectives Understand how variability in arrivals and services can lead to a build up of queues. ........................................................................................................................................................................................................................................ ........................................................................................................................................................................................................................................ ........................................................................................................................................................................................................................................

Agenda Review: General Queuing Models Special Queuing Models SofOptics

Agenda Review: General Queuing Models Special Queuing Models

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)

Solution: Calling Center ........................................................................................................................................................................................................................................ ........................................................................................................................................................................................................................................ ........................................................................................................................................................................................................................................ Source: Managing Business Process Flows (1999)

Cost of Service Tradeoff Total cost Expected costs (Service) Capacity Cost (Customer) Waiting Cost Level of service

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. 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)

Solution: Ticket Outlet ........................................................................................................................................................................................................................................ ........................................................................................................................................................................................................................................ ........................................................................................................................................................................................................................................ Source: Managing Business Process Flows (1999)

Is Idleness a Good Thing? Base (minimum) Capacity needed: R Safety Capacity: Rp - R capacity carried in excess of expected demand to cover for system variability Idle Percentage: 1 - ρ = 1 - R/Rp Source: Managing Business Process Flows (1999)

Performance Measures to Focus on Sales Throughput (R) Abandoning rate (Ra ) Cost Capacity utilization (r) 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)

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)

Example: Calling Center with 2 CSRs (2 servers) 2 phone numbers 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 1 phone number: pooling Both CSRs share the same telephone number and the customers on hold are in a single queue Server Queue 50% Servers Queue ........................................................................................................................................................................................................................................ ........................................................................................................................................................................................................................................ ........................................................................................................................................................................................................................................ Source: Managing Business Process Flows (1999)

Solution: Calling Center with 2 CSRs (servers) ........................................................................................................................................................................................................................................ ........................................................................................................................................................................................................................................ ........................................................................................................................................................................................................................................ Source: Managing Business Process Flows (1999)

Agenda Review: General Queuing Models Special Queuing Models SofOptics