Chapter 18 Management of Waiting Lines Cheng Li, Ph.D. Management Department California State University, Los Angeles

Waiting Lines Examples of waiting lines Car wash: three stages, multiple channels (servers) for vacuuming and drying, single channel for washing A network printer: single stage, single channel A maintenance worker: single channel, finite number of customers Raw materials waiting to be processed

Elements of Queuing System: single phase, single channel Arrivals Service Waiting line Exit Processing order System

Elements of Queuing System: multiple phases, multiple channels, single line

Waiting Lines Queuing theory: mathematical analysis of waiting lines Decision: capacity level that optimally balances the tradeoff between: Customer waiting costs, and Service capacity costs

Costs of Waiting Lines Waiting lines: delay in work process (“inventory” holding cost) Loss of business: customers refusing to wait Loss of goodwill, decrease in customer satisfaction Impact on other operations in the same process: disruptions caused by congestion at the bottleneck Cost of waiting space

Cost Tradeoff Cost Service capacity Total cost Customer waiting cost = + Total cost Lowest TC Cost of service capacity Cost of customers waiting Service capacity

Waiting Lines Assumption: no change in process design, but future improvement may benefit from the results of analysis Roles of Queuing Analysis: Estimate system performance for given capacity decisions Provide input for future process improvement

System Characteristics Customer population: Infinite source: unlimited number of potential arrivals in a given period of time Finite source: limited number of potential arrivals in a given period of time Number of phases: distinguished by: a designated server for each phase a potential waiting line between consecutive phases

System Characteristics Number of channels (servers): max. number of customers being served at the same time Number of lines (multiple channels) Single line vs. Multiple lines Advantages of single lines Less waiting Higher utilization of capacity Customer perception Question: Why do some businesses still use multiple lines?

System Characteristics Arrival patterns Arrival rate: number of arrivals per period probabilistic: Poisson distribution deterministic: predetermined (appointment system) Inter-arrival time: time between two consecutive arrivals, inverse of arrival rate Service patterns: probabilistic vs. deterministic Service Rate: inverse of service time Service Time: Probabilistic: exponential distribution Deterministic: constant

Poisson & Exponential Distributions Poisson: e.g. service rate Exponential: e.g. service time Poisson: mean = 4 (e.g. 4 customers per hour) Exponential: lamda = 1 (parameter value) Service time for the exponential distribution is inverse of service rate values. For instance: 2 customers per hour => 0.5 hours of service time per customer; 4 hours of service time per customer => 0.25 customers per hour.

System Characteristics Queue discipline: priority rules used to determine the order of customers FCFS: First Come, First Server EDD: Earliest Due Date SPT: Shortest Processing Time Appointment system

System Performance: Basic Measures Capacity measures Capacity utilization Average number of customers waiting Average number of customers in system (waiting + being served) Customer measures Average time customers wait Average service time Average time in system

Performance Measurement for Queuing Models: Infinite Source Single channel, exponential service time Single channel, constant service time Multiple channel, exponential service time Other assumptions: Single phase Poisson arrival distribution (= exponential distribution for inter-arrival time) FCFS

Infinite Source: General Formulas System Utilization Idle Rate Avg. # of customers in line Avg. # of customers in system Avg. waiting time Avg. service time Avg. time in system

Infinite Source: single channel Avg. # of customers in line Prob.{0 customer in system} Prob.{n customers in system} Prob.{customers in system<n}

Infinite Source: constant service time Avg. # of customers in line Eliminate variability in service time Reduce waiting line/time by half

System Performance: Other Measures Question: Is “average” a good measure? HALF of the total is worse than average (for a symmetric distribution) Does not tell about the worst cases Does not tell about spread of distribution SD=2: 12.1% SD=1: 5.4%

Infinite Source: single channel Probability regarding # of customers in system:

Infinite Source: single channel Probability regarding # of customers in system:

System Performance: Application of Measures Probability that an arrival will have to wait (1-P0) % of callers who are put on hold Probability of more than a given number of customers in line % of callers who have to wait for more than a given amount of time Probability that number of customers waiting will exceed a given number Chance of exceeding the capacity limit of the waiting area

System Performance Implied cost Implied constraints Capacity costs: e.g. #servers x wage Customer related costs: e.g. costs associated with a given level of waiting Space: unlimited space assumed, but: “Will space limit be exceeded?” Implied constraints Business process: how service is divided into phases, tasks at each phase, etc.

Utilization & Waiting: 20 sec@25% 60 sec@50%

Queuing Models Single channel, exponential service time Single channel, constant service time Multiple channel, exponential service time Multiple priority service

Priority Model Arrivals Service Waiting line Exit Processing order System 1 2 3 Arrivals are assigned a priority as they arrive

Finite-Source Formulas Table 19-6 Average number being served Service factor Average number waiting Average waiting time Average number running Number in population

Finite-Source Queuing Not waiting or being served Waiting Being served J L H U W T