Post on 13-Apr-2015
description
Waiting Line Models
(Queuing Theory)
© 2012 Lew Hofmann
Lines in Operations Management
• Assembly lines
• Production lines
• Trucks waiting to unload or load
• Workers waiting for parts
• Customers waiting for products
• Broken equipment waiting to be fixed
• Customers waiting for service
© 2012 Lew Hofmann
Costs The cost of waiting
Losing customers because of long lines• Reneging: Customers get tired of waiting and leave
• Balking: Customers see a long line and don’t get in line.
Unusable (idle) equipment awaiting repairs• EG: Broken assembly line machinery.
The cost of service Paying people to provide service to customers Customers can be people, machines, or other objects needing service.
© 2012 Lew Hofmann
Cost of Providing Service
• Paying repairmen to fix broken machines
• Paying dock workers to load and unload trucks
• Paying customer-service people
• Using more production people to speed up the line
• Leasing of service equipment and facilities
• Paying checkout cashiers
© 2012 Lew Hofmann
Queuing System Costs
Number of Servers
Costs
Cost of Servicegoes up as you pay for more servers.
Costs of Waitinggoes down as service improves.
Total Cost
Optimal # of servers
Note that the lowest cost system requires some customer waiting.
Fewer servers often means longer waiting for customers. Many servers
means little or no waiting, but higher service costs.
© 2012 Lew Hofmann
Queuing theory in quantitative technique is useful for determining the optimum number of service facilities
Managerial application of queuing theory
1. Aircraft at landing and take-off from busy airport.
2. Routing sales persons 3. Jobs in production control
© 2012 Lew Hofmann
What Queuing Models Tell Us.
• Average time in line for a customer.
• Average number of customers in line.
• Average time in the system for a customer.
• Average number of customers in the system at any time.
• Probability of n number of customers in the system at any given time.
• NOTE: “In The System” includes customers who are in line plus the customers being served.
© 2012 Lew Hofmann
ARRIVAL SYSTEM(How customers arrive)
QUEUE(The nature of the waiting line or lines of customers)
The Waiting Line System
SERVICE FACILITY(How customers progress through the service facility)
© 2012 Lew Hofmann
Waiting Line Models
Served customers
Arrival System
Service System
Waiting line (Queue)
Priority rule
Service facilities
The sequence in which customers are admitted into the service facility.
Population
© 2012 Lew Hofmann
Arrival System
• Arrival Populations are either…
• Limited (EG: Counters for handicapped persons.)
• Unlimited (EG: cars arriving at a toll booth)
• Arrival Patterns are either…
• Random (Each arrival is independent e.g.in Bank)
• Scheduled (EG: Doctor’s office visits)
• Behavior of the Arrivals
• Balking (Seeing a long line and avoiding it.)
• Reneging (Get tired of waiting and leave the line)
• Jockeying (Switching lines)
© 2012 Lew Hofmann
The Queue
• Queue Length is either..
• Unlimited (EG: cars in line at a toll booth)
• Limited (Finite) EG: # of e-mail messages allowed.
• Queue Discipline (order of service)
• FIFO (First-In, First-Out)
• LIFO (Last-In, First-Out)
• Priority
© 2012 Lew Hofmann
The Service Facility
• Channels• How many paths (ways to get through the
system) are there after getting in line?• In Bank Customer can use
• Channel I: ATM machine to withdraw cash
• Channel II:May stand in queue to get it from cashier.
© 2012 Lew Hofmann
Single-channel, Single-phaseOne way through the system
and one stop for service
Service Facility
© 2012 Lew Hofmann
Multi-channel, Single-phase
Service Facility
Service Facility
Once in line, you have at least two choices of how to get through the system, but only one stop.
© 2012 Lew Hofmann
Multiple v.s. Single Customer Queue Configuration
1. The service provided can be differentiated Ex. Supermarket express lanes
2. Labor specialization possible
3. Customer has more flexibility
4. Balking behavior may be deterredSeveral medium-length
lines are less intimidating than one very long line
1. Guarantees fairness FIFO applied to all arrivals
2. No customer anxiety regarding choice of queue
3. Avoids “cutting in” problems
4. The most efficient set up for minimizing time in the queue
5. Jockeying (line switching) is avoided
Multiple Line Advantages Single Line Advantages
© 2012 Lew Hofmann
Components of a Basic Queuing Process The Service Mechanism
Can involve one or several service facilities with one or several parallel service channels (servers) - Specification is required
The service provided by a server is characterized by its service time• Specification is required and typically involves data
gathering and statistical analysis.• Most analytical queuing models are based on the
assumption of exponentially distributed service times, with some generalizations.
The queue discipline Specifies the order by which jobs in the queue are being
served. Most commonly used principle is FIFO. Other rules are, for example, LIFO, SPT, EDD… Can entail prioritization based on customer type.
© 2012 Lew Hofmann
n = total number of customers in the system at time ‘t’
l = average number of customers arriving per unit time
µ = average number of customers being serviced per unit time
C = number of parallel service channels
Ls= average number of customers in the system = /(µ - )
Lq = average number of customers waiting in the queue = x / µ(µ - )
NOTATIONS : (M/M/1) [ follows Poisson Distribution and µ follows Exponential Distribution ]
© 2012 Lew Hofmann
NOTATIONS :
Ws = average number of time of customer in the system in both waiting and in service. = 1/ (µ - )
Wq = average waiting time of customer in a queue = / µ(µ - )
Pn(t) = Prob. That there are n customers in the system at any time t both waiting and in service= ( 1- / µ) ( / µ)^ n
Utilization factor = [Average service completion time(1/ µ)]/ [Average inter-arrival time (1/ )] P[system is busy] = = / µ = Expected number of customers in the systemP[ system is idle] = 1- P[ system is busy] = 1 -
© 2012 Lew Hofmann