Waiting Lines and Queuing Models

Queuing Theory  The study of the behavior of waiting lines Importance to business There is a tradeoff between faster lines and increased costs  faster lines suggests an increase in service, thus an increase in costs  longer waiting times negatively affects customer satisfaction What is the ‘ideal’ level of services that a firm should provide?

Management Uses from Queuing Theory  Is it worthwhile to invest effort in reducing the service time?  How many servers should be employed?  Should priorities for certain types of customers be introduced?  Is the waiting area for customers adequate?  Answers to these questions can be obtained with Analytic methods or queuing theory (formula based); and Simulation (computer based).

Queuing System Characteristics  Arrivals  Waiting in Line  Service Facility

Arrival Characteristics  Size of the calling population Finite: ex. 300 computers on campus maintained by 5 computer technicians (customers arriving for service are limited) Infinite: ex. cars arriving at a highway tollbooth, shoppers arriving at a supermarket (the source is forever “abundant”)  Pattern of Arrivals Nonrandom: arrivals take place according to some known schedule (ex. assembly line) Random: arrivals are independent and cannot be predicted exactly

Random Pattern of Arrival  Poisson Distribution a probability distribution that can be used to determine the probability of X transactions arriving in a given time interval P(X) = for X = 0, 1, 2, 3, 4 where,P(X) = probability of X arrivals X = number of arrivals per unit of time = average arrival rate e = e - X X!

Examples of Poisson Distribution for Arrival Times = 2 Distribution Probability = 4 Distribution X X

Arrival Characteristics  Size of the calling population Finite: ex. 300 computers on campus maintained by 5 computer technicians Infinite: ex. Cars arriving at a highway tollbooth, shoppers arriving at a supermarket  Pattern of Arrivals Random: arrivals are independent and cannot be predicted exactly Nonrandom: arrivals take place according to some known schedule  Behavior of the Arrivals Balking: customers who refuse to enter the system because the line is too long Reneging: customers who enter the queue but leave without completing their transactions Jockeying: switching between lines

Waiting Line Characteristics  Queue Length Limited: the length of the queue is limited by physical restrictions ex. waiting room Unlimited: the length of the queue is not restricted  Queue Discipline Rule by which customers in the line are to receive service Static: FCFS, first come first serve, FIFO, first in first out Dynamic: Priority e.g., rush jobs at a shop are processed ahead of regular jobs

Service Facility Characteristics  Basic Queuing System Configurations Single Channel one service provider per phase Multiple Channel more than one service provider in a phase

Basic Single Queue Configurations Service Facility Arrivals Queue Departures after Service Single-Channel, Single-Phase System Type 1 Service Facility Arrivals Queue Departures after Service Single-Channel, Multiphase System Type 2 Service Facility Arrivals Queue Departures after Service Multichannel, Single-Phase System Service Facility 1 Service Facility 2 Service Facility 3 Arrivals Queue Departures after Service Multichannel, Multiphase System Type 1 Service Facility 1 Type 1 Service Facility 2 Type2 Service Facility 1 Type 2 Service Facility 2

Multiple Queue Configurations Arrivals Departures after Service Multiple Queue Service Facility 1 Service Facility 2 Service Facility 3 Arrivals Take a Number Service Facility 4 73 Departures after Service Service Facility 1 Service Facility 2 Service Facility 3 Service Facility

Service Facility Characteristics  Basic Queuing System Configurations Single Channel one service provider per phase Multiple Channel more than one service provider in a phase  Service Time Distribution Constant: it takes the same amount of time to service each customer or unit Random: service times vary across customers or units

Examples of Exponential Distribution for Service Times Service Time (Minutes) Probability (Service Takes Longer Than X Minutes) = e -uX for X > 0 Probability (for intervals of 1 minute) u = Average Number Served per Minute Average Service Time of 20 Minutes Average Service Time of 1 hour

Assumptions of the Single- Channel, Single-Phase Model  Arrivals are served on a FIFO basis  Every arrival waits to be served regardless of the length of the line: that is there is no balking or reneging  Arrivals are independent of preceding arrivals, but the average number of arrivals (the arrival rate, λ) does not change over time  Arrivals are described by a Poisson probability distribution and come from an infinite or very large population  Service times also vary from one customer to the next and are independent of one another, but their average rate (μ) is known  Service times occur according to the negative exponential probability distribution  The average service rate is greater than the average arrival rate

Idea of Uncertainty  Note here that integral to queuing situations is the idea of uncertainty in Interarrival times (arrival of customers) Service times (service time per customer) This means that probability and statistics are needed to analyze queuing situations.

System Performance Measures  Important to measuring the performance of the system are the parameters: λ = the average number of arrivals per time period μ = the average number of people or items served per time period

System Performance Measures  Number of units in the system (customers) Average number in system (L or L s ) Average queue length (L q )  Waiting Times Average time in the system (W or W s ) Average time in queue (W q )  Utilization Rates Utilization factor () Probability of idle time (P 0 )

Queuing Equations Average number in system (L or L s ) Average queue length (L q ) Average time in the system (W or W s ) Average time in queue (W q ) Utilization factor () Probability of idle time (P 0 ) L= L q = W= W q = = P 0 = 1 - λ μ - λ λ 2 μ (μ – λ) 1 μ - λ λ μ (μ – λ) λμλμ λμλμ

Queuing Equations Probability that the number of customers in the system is greater than k, P n>k where n = number of units in the system P n>k = ( ) λ k + 1 μ

When to use what model?  Use Single-channel model, when you have Only one service provider Infinite source (calling population) Random pattern of arrivals (Pois Dist) No balking, reneging, jockeying Random (inconstant) service times (Expo Dist) FIFO

When to use what model? (2)  Use Multi-channel model, when you have More than one service providers Infinite source (calling population) Random pattern of arrivals (Pois Dist) No balking, reneging, jockeying Random (inconstant) service times (Expo Dist) but both channel must perform at the same rate

When to use what model? (3)  Use Constant-service time model, when you have Constant service times (a fixed cycle) Infinite source (calling population) Random pattern of arrivals (Pois Dist) No balking, reneging, jockeying  The question will be asking about either to choose the new or the old machines.

When to use what model? (4)  Use finite population model, when you have Finite source (calling population) Random (inconstant) service times (Expo Dist) Only one service providers Random pattern of arrivals (Pois Dist) FCFS

SUM Finite Finite pop model Yes No Constant ServTime 1 Channel>1 Channel Yes Constant Model No Single-Chn ModelMulti-Chn Model