Module D Waiting Line Models

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

Components of Queuing System Source of customers— calling population Server Arrivals Waiting Line or “Queue” Served customers

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

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

Elements of a Waiting Line Queue discipline Order in which customers are served First come, first served is most common Length can be infinite or finite Infinite is most common Finite is limited by some physical structure (not covered)

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

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

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

Operating Characteristics Mathematics of queuing theory does not provide optimal or best solutions 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

Operating Characteristics NOTATION OPERATING CHARACTERISTIC L Average number of customers in the system (waiting and being served) Lq Average number of customers in the waiting line W Average time a customer spends in the system (waiting and being served) Wq Average time a customer spends waiting in line

Operating Characteristics NOTATION OPERATING CHARACTERISTIC P0 Probability of no (zero) customers in the system Pn Probability of n customers in the system  Utilization rate; the proportion of time the system is in use

Single-Channel, Single-Phase Models All assume Poisson arrival rate Variations Exponential service times Constant service times Others are not considered

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

Formulas for Single-Server Model Probability that no customers are in the system (either in the queue or being served) P0 = 1 -   Probability of exactly n customers in the system Pn = • P0 n   = 1 - L =   -  Average number of customers in the system Average number of customers in the waiting line Lq =  ( - )

Formulas for Single-Server Model Average time a customer spends in the queuing system W = = 1  - L  Average time a customer spends waiting in line to be served Wq =  ( - )  =   Probability that the server is busy and the customer has to wait Probability that the server is idle and a customer can be served I = 1 -    = 1 - = P0

A Single-Server Model Given  = 24 per hour,  = 30 customers per hour Probability of no customers in the system P0 = 1 - = 1 - = 0.20   24 30 L = = = 4 Average number of customers in the system   -  24 30 - 24 Average number of customers waiting in line Lq = = = 3.2 (24)2 30(30 - 24) 2 ( - )

A Single-Server Model Given  = 24 per hour,  = 30 customers per hour Average time in the system per customer W = = = 0.167 hour 1  - 30 - 24 Average time waiting in line per customer  (-) 24 30(30 - 24) Wq = = = 0.133 Probability that the server will be busy and the customer must wait  = = = 0.80   24 30 Probability the server will be idle I = 1 -  = 1 - 0.80 = 0.20

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

Waiting Line Cost Analysis Add extra employee to increase service rate from 30 to 40 customers per hour Extra employee costs $150/week Each one-minute reduction in customer waiting time avoids $75 in lost sales Waiting time with one employee = 8 minutes Wq = 0.038 hours = 2.25 minutes 8.00 - 2.25 = 5.75 minutes reduction 5.75 x $75/minute/week = $431.25 per week New employee saves $431.25 - $150.00 = $281.25/wk

Waiting Line Cost Analysis New counter costs $6000 plus $200 per week for checker Customers divide themselves between two checkout lines Arrival rate is reduced from = 24 to = 12 Service rate for each checker is  = 30 Wq = 0.022 hours = 1.33 minutes 8.00 - 1.33 = 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 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 would take up potentially valuable floor space

Constant Service Times 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

Operating Characteristics for Constant Service Times   P0 = 1 - Probability that no customers are in system Average number of customers in queue Lq = 2 2( - ) Average number of customers in system L = Lq +  

Operating Characteristics for Constant Service Times Average time customer spends in queue Wq =  Lq  Average time customer spends in the system W = Wq + 1     = Probability that the server is busy

Constant Service Times Automated car wash with service time = 4.5 min Cars arrive at rate  = 10/hour (Poisson)  = 60/4.5 = 13.3/hour 2 2( - ) (10)2 2(13.3)(13.3 - 10) Lq = = = 1.14 cars waiting Wq = = 1.14/10 = .114 hour or 6.84 minutes Lq 

Multiple-Channel, Single-Phase Models Two or more independent servers serve a single waiting line Poisson arrivals, exponential service, infinite calling population s> P0 = 1 s!   s s s -   n=s-1 n=0 n! n +

Multiple-Channel, Single-Phase Models Computing P0 can be time-consuming. Tables can used to find P0 for selected values of  and s. Two or more independent servers serve a single waiting line Poisson arrivals, exponential service, infinite calling population s> P0 = 1 s!   s s s -   n=s-1 n=0 n! n +

Multiple-Channel, Single-Phase Models Probability of exactly n customers in the system Pn = P0, for n > s 1 s! sn-s   n P0, for n > s n! Probability an arriving customer must wait Pw = P0 1 s! s s -  s   Average number of customers in system L = P0 + (/)s (s - 1)!(s - )2  

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