Waiting Line Analysis OPIM 310-Lecture 3 Instructor: Jose Cruz

Elements of Waiting Line Analysis Queue Queue A single waiting line A single waiting line Waiting line system consists of Waiting line system consists of Arrivals Arrivals Servers Servers Waiting line structures Waiting line structures

Common Queuing Situations Dock workers load and unloadShips and bargesHarbor Repair people fix machinesBroken machinesMachine maintenance Transactions handled by tellerCustomerBank Switching equipment to forward calls CallersTelephone company Computer processes jobsPrograms to be runComputer system Treatment by doctors and nurses PatientsDoctor’s office Collection of tolls at boothAutomobilesHighway toll booth Checkout clerks at cash register Grocery shoppersSupermarket Service ProcessArrivals in QueueSituation

Components of Queuing System Source of customers— calling population ServerArrivals Waiting Line or“Queue”Servedcustomers

Parts of a Waiting Line Figure D.1 Dave’s Car Wash enterexit Population of dirty cars Arrivals from the general population … Queue (waiting line) Servicefacility Exit the system Arrivals to the system Exit the system In the system Arrival Characteristics  Size of the population  Behavior of arrivals  Statistical distribution of arrivals Waiting Line Characteristics  Limited vs. unlimited  Queue discipline Service Characteristics  Service design  Statistical distribution of service

Elements of a Waiting Line Calling population Calling population Source of customers Source of customers Infinite - large enough that one more customer can always arrive to be served Infinite - large enough that one more customer can always arrive to be served Finite - countable number of potential customers Finite - countable number of potential customers Arrival rate ( ) Arrival rate ( ) Frequency of customer arrivals at waiting line system Frequency of customer arrivals at waiting line system Typically follows Poisson distribution Typically follows Poisson distribution

Elements of a Waiting Line Service time Service time Often follows negative exponential distribution Often follows negative exponential distribution Average service rate =  Average service rate =  Arrival rate ( ) must be less than service rate  or system never clears out Arrival rate ( ) must be less than service rate  or system never clears out

Distribution Of Arrivals Assumption: arrivals occur randomly and independently on each otherAssumption: arrivals occur randomly and independently on each other Poisson distribution provides a good description of the arrival pattern:Poisson distribution provides a good description of the arrival pattern: P(x) = for x = 0, 1, 2, 3, 4, … e - x x! whereP(x)=probability of x arrivals x=number of arrivals per unit of time =average arrival rate =average arrival rate e= (which is the base of the natural logarithms)

Poisson Distribution Probability = P(x) = e - x x! – – – – – – Probability Distribution for = 2 x – – – – – – Probability Distribution for = 4 x 1011

Distribution Of Service Times In general service time can follow any arbitrary distribution The simplest, however, is an exponential:

Distribution Of Service Times – – – – – – – – – – – Probability that service time ≥ 1 ||||||||||||| Time t in hours Probability that service time is greater than t = e -µt for t ≥ 1 µ = Average service rate e = Average service rate (µ) = 1 customer per hour Average service rate (µ) = 3 customers per hour  Average service time = 20 minutes per customer

Elements of a Waiting Line Queue discipline Queue discipline Order in which customers are served Order in which customers are served First come, first served is most common First come, first served is most common Length can be infinite or finite Length can be infinite or finite Infinite is most common Infinite is most common Finite is limited by some physical structure Finite is limited by some physical structure

Basic Waiting Line Structures Channels are the number of parallel servers Channels are the number of parallel servers Phases denote number of sequential servers the customer must go through Phases denote number of sequential servers the customer must go through

Single-Channel Structures Single-channel, single-phase Waiting lineServer Single-channel, multiple phases Servers Waiting line

Multi-Channel Structures Servers Multiple-channel, single phase Waiting line Servers Multiple-channel, multiple-phase

Operating Characteristics Mathematics of queuing theory does not provide optimal or best solutions Mathematics of queuing theory does not provide optimal or best solutions Operating characteristics are computed that describe system performance Operating characteristics are computed that describe system performance Steady state is constant, average value for performance characteristics that the system will reach after a long time Steady state is constant, average value for performance characteristics that the system will reach after a long time

Operating Characteristics NOTATIONOPERATING CHARACTERISTIC LAverage number of customers in the system (waiting and being served) L q Average number of customers in the waiting line WAverage time a customer spends in the system (waiting and being served) W q Average time a customer spends waiting in line

Operating Characteristics NOTATIONOPERATING CHARACTERISTIC P 0 Probability of no (zero) customers in the system P n Probability of n customers in the system  Utilization rate; the proportion of time the system is in use

Cost Relationship in Waiting Line Analysis Expected costs Level of service Total cost Service cost Waiting Costs

Waiting Line Costs and Quality Service Traditional view is that the level of service should coincide with minimum point on total cost curve Traditional view is that the level of service should coincide with minimum point on total cost curve TQM approach is that absolute quality service will be the most cost- effective in the long run TQM approach is that absolute quality service will be the most cost- effective in the long run

Single-Channel, Single- Phase Models All assume Poisson arrival rate All assume Poisson arrival rate Variations Variations Exponential service times Exponential service times General (or unknown) distribution of service times General (or unknown) distribution of service times Constant service times Constant service times Exponential service times with finite queue length Exponential service times with finite queue length Exponential service times with finite calling population Exponential service times with finite calling population

Basic Single-Server Model Assumptions: Assumptions: Poisson arrival rate Poisson arrival rate Exponential service times Exponential service times First-come, first-served queue discipline First-come, first-served queue discipline Infinite queue length Infinite queue length Infinite calling population Infinite calling population = mean arrival rate = mean arrival rate  = mean service rate  = mean service rate

Formulas for Single- Server Model L =   -   - Average number of customers in the system Probability that no customers are in the system (either in the queue or being served) P 0 = 1 -  Probability of exactly n customers in the system P n = P 0 n = 1 -  n  Average number of customers in the waiting line L q =   (  - )

Formulas for Single- Server Model  =  Probability that the server is busy and the customer has to wait Average time a customer spends in the queuing system W = = 1  -  L Probability that the server is idle and a customer can be served I = 1 -   = 1 - = P 0 Average time a customer spends waiting in line to be served W q =  (  - )

A Single-Server Model Given = 24 per hour,  = 30 customers per hour Probability of no customers in the system P 0 = 1 - = 1 - = 0.20  2430 L = = = 4 Average number of customers in the system  -  Average number of customers waiting in line L q = = = 3.2 (24) 2 30( ) 2  (  - )

A Single-Server Model Given = 24 per hour,  = 30 customers per hour Average time in the system per customer W = = = hour 1  -  Average time waiting in line per customer W q = = =  (  -  ) 24 30( ) Probability that the server will be busy and the customer must wait  = = = 0.80  2430 Probability the server will be idle I = 1 -  = = 0.20

1.Another employee to pack up purchases 2.Another checkout counter Waiting Line Cost Analysis To improve customer services management wants to test two alternatives to reduce customer waiting time:

Waiting Line Cost Analysis Add extra employee to increase service rate from 30 to 40 customers per hour Add extra employee to increase service rate from 30 to 40 customers per hour Extra employee costs $150/week Extra employee costs $150/week Each one-minute reduction in customer waiting time avoids $75 in lost sales Each one-minute reduction in customer waiting time avoids $75 in lost sales Waiting time with one employee = 8 minutes Waiting time with one employee = 8 minutes Example 2 W q = hours = 2.25 minutes = 5.75 minutes reduction 5.75 x $75/minute/week = $ per week New employee saves $ $ = $281.25/wk

Waiting Line Cost Analysis New counter costs $6000 plus $200 per week for checker New counter costs $6000 plus $200 per week for checker Customers divide themselves between two checkout lines Customers divide themselves between two checkout lines Arrival rate is reduced from  = 24 to  = 12 Arrival rate is reduced from  = 24 to  = 12 Service rate for each checker is  = 30 Service rate for each checker is  = 30 Example 2 W q = hours = 1.33 minutes = 6.67 minutes 6.67 x $75/minute/week = $500.00/wk - $200 = $300/wk Counter is paid off in 6000/300 = 20 weeks

Waiting Line Cost Analysis Adding an employee results in savings and improved customer service Adding an employee results in savings and improved customer service Adding a new counter results in slightly greater savings and improved customer service, but only after the initial investment has been recovered Adding a new counter results in slightly greater savings and improved customer service, but only after the initial investment has been recovered A new counter results in more idle time for employees A new counter results in more idle time for employees A new counter would take up potentially valuable floor space A new counter would take up potentially valuable floor space Example 2

Constant Service Times Constant service times occur with machinery and automated equipment Constant service times occur with machinery and automated equipment Constant service times are a special case of the single-server model with general or undefined service times Constant service times are a special case of the single-server model with general or undefined service times

Finite Queue Length A physical limit exists on length of queue A physical limit exists on length of queue M = maximum number in queue M = maximum number in queue Service rate does not have to exceed arrival rate (  ) to obtain steady-state conditions Service rate does not have to exceed arrival rate (  ) to obtain steady-state conditions

Single-Channel Waiting Line Model With Poisson Arrivals And Arbitrary Service Times (M/G/1) Notation:

Single-Channel Waiting Line Model With Poisson Arrivals And Arbitrary Service Times (M/G/1) Operating Characteristics The probability that no units are in the system The average number of units in the waiting line The average number of units in the system

Single-Channel Waiting Line Model With Poisson Arrivals And Arbitrary Service Times (M/G/1) Operating Characteristics The average time a unit spends in the waiting line The average time a unit spends in the system The probability that an arriving unit has to wait for service The average number of units in the waiting line

Multiple-Channel, Single-Phase Models Two or more independent servers serve a single waiting line Two or more independent servers serve a single waiting line Poisson arrivals, exponential service, infinite calling population Poisson arrivals, exponential service, infinite calling population s  >  s  >  P 0 = 11s!  s s  s  - s  -  n=s-1 n=0 1 11n!n!11n!n!  n +

Multiple-Channel, Single-Phase Models Two or more independent servers serve a single waiting line Two or more independent servers serve a single waiting line Poisson arrivals, exponential service, infinite calling population Poisson arrivals, exponential service, infinite calling population s  >  s  >  P 0 = 11s!  s s  s  - s  -  n=s-1 n=0 1 11n!n!11n!n!  n + Computing P 0 can be time-consuming. Tables can used to find P 0 for selected values of  and s.

Multiple-Channel, Single-Phase Models Probability of exactly n customers in the system P n = P 0, for n > s 1 s! s n-s n P 0, for n > s 1 11n!n!11n!n!  n Probability an arriving customer must wait P w = P s!s!11s!s! s  s  - s  - s Average number of customers in system L = P 0 +  ( /  ) s (s - 1)!(s  - ) 2 

Multiple-Channel, Single-Phase Models W = L Average time customer spends in system   =  /s  Utilization factor Average time customer spends in queue W q = W - = 1 L q L q = L -  Average number of customers in queue

Multiple-Server System Customer service area = 10 customers/area = 10 customers/area  = 4 customers/hour per service rep s  = (3)(4) = 12 P 0 = Probability no customers are in the system Number of customers in the service department L = 6 Waiting time in the service department W = L / = 0.60

Multiple-Server System Customer service area = 10 customers/area = 10 customers/area  = 4 customers/hour per service rep s  = (3)(4) = 12 L q = L - /  = 3.5 Number of customers waiting to be served Average time customers will wait in line W q = L q / = 0.35 hours Probability that customers must wait P w = 0.703

Add a 4th server to improve service Add a 4th server to improve service Recompute operating characteristics Recompute operating characteristics P 0 = prob of no customers P 0 = prob of no customers L = 3.0 customers L = 3.0 customers W = 0.30 hour, 18 min in service W = 0.30 hour, 18 min in service L q = 0.5 customers waiting L q = 0.5 customers waiting W q = 0.05 hours, 3 min waiting, versus 21 earlier W q = 0.05 hours, 3 min waiting, versus 21 earlier P w = 0.31 prob that customer must wait P w = 0.31 prob that customer must wait Improving Service

Splitting Arrival Flow Arrival rate sometimes depends on type of customer Or, some customers prefer one queue over another (when there is a choice) Idea: determine percentage of customers joining a queue based on type of preference

Splitting Flow Example Queue system with two lines Line 1 is served by 2 clerks, each clerk has an average service time of 5 minutes Line 2 is served by a single automated system that takes 2 minutes on average 75% of the customers prefer the line served by the human clerks What is the waiting time, system size, etc?

Cost Evaluation Service Cost = (Number of servers) x (wages per time unit) = s C s Waiting Cost = (Number of customers waiting in the system) x (cost of waiting per time unit) = L C w Total cost = service cost + waiting cost

Decision Areas  Arrival Rates  Number of Service Facilities  Number of Phases  Number of Servers Per Facility  Server Efficiency  Priority Rule  Line Arrangement