Waiting line Models

Some ExamplesSituation Arriving customers Service facility

Sale of Theatre tickets Theatre goers Ticket booking windows

Banking Transactions Bank patrons Bank Tellers

Arrival of trucks to carry materials

Trucks Loading crews and facilities

Sale of Railway tickets Passengers Ticket counters

Transfer of e-messages E-messages Transmission line

Flow of ships to seashore Ships Docking facility

Billing in supermarket Shoppers Checkout counters

Goal of Waiting line models

• Why there is a waiting line?• What are the cost associated with waiting?

The goal is to provide an economic balance between cost of providing service and the cost of customers waiting

Structure of Queuing system

• Population may be finite or may be infinite• Arrivals follow Poisson’s distribution• Services follow Exponential distribution

Arrival process1. According to source: Finite or Infinite2. According to number: single or group

Poisson’s Distribution for

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 160

0.05

0.1

0.15

0.2

0.25

Series1

3, 0,1,2...15x

Understanding use of Poisson’s distribution

Example: On an average 5 customers reach a Saloon every hour. Determine the probability that less than 2 customers will reach in 30 minutes period, assuming that the arrival follow Poisson’s distribution. What will be the probability that exactly 5 customers will reach in an hour?

Poisson’s function: ( ) , 0,1,2...

!

xef x x

x

Different Structures of Service system

• Speed of Service:In an waiting line system, the speed with which service is provided can be expressed as either a) Service rate : (10 customers / hour)b) Service time: ( 6minutes / customer)

• Service follows exponential distribution with pdf:

where is the mean number of occurrences of an event per time unit

• Queue Structure:a) FCFSb) LCFS

( ) , , 0tf t e t

Understanding use of Exponential distribution

( ) 1 aP t a e ( ) bP t b e

( ) a bP a t b e e

• Example on Exponential distribution:The manager of a bank observes that 18 customers are served by a cashier in an hour. Assuming the service time has an exponential distribution, what is the probability that

a) A customer shall be free within 3 minutesb) A customer shall be serviced in more than 12 minutes?

Operating Characteristics= Arrival rate

µ = Service rate

Lq = the average number waiting for service

L = the average number in the system (i.e.,waiting for service or being served)

P0 = the probability of zero units in the system

p = = the system utilization (percentage of time servers are busy serving customers)

Wq = the average time customers must wait for service

W = the average time customers spend in the system (i.e., waiting for service and service time)

/

Model 1: M/M/1/∞/ ∞

• A single-server model is appropriate when these conditions exist:– A Poisson arrival rate.

– A negative exponential service time.

– One server or channel

– First-come, first-served processing order.

– An infinite population.

– No limit on queue length i.e. infinite queue length

Relationship between average number and average time

Formulas for basic model

Formulas for basic model

Values for Lq and P0 given λ ⁄ μ and s

Example:

• Arrivals at a patient at a registration counter are considered to follow Poisson’s distribution, with an average time of 3 minutes between one arrival and the next. The time taken by the staff at registration desk is assumed to be distributed exponentially with a mean of 2 minutes. Find

• Number of patients on average waiting in the queue• Average waiting time of a patient in the queue or in

completing registration• Average idle time for the staff in a day ( 8 hours shift)

Class exercise:1. A T.V repairman finds that the time spent on his jobs has an

exponential distribution with mean 30 minutes. If he repairs sets in the order in which they came in and if the arrival of sets follow Poisson with an average rate of 10 per 8-hour day, what is his idle time each day? how many jobs are ahead of the set just brought in?

2. In a railway yard, goods train arrive at the rate of 30 per day. Assuming that the arrival follow Poisson and the service time follow exponential with an average of 36 minutes, find a) the queue lengthb) Probability that the queue size exceeds 10c) If the arrival rate increases to 33 per day, what will be the expected queue length?

Model 2: M/M/s/∞/ ∞

• A multiple-server model is appropriate when these conditions exist:– A Poisson arrival rate.

– A negative exponential service time.

– multiple servers or channels (denoted by s)

– First-come, first-served processing order.

– An infinite population.

– No limit on queue length i.e. infinite queue length

– The same mean service rate for all servers

Formulas for multiple server model

s

Combined service rate

Infinite Source Values for Lq and P0 given λ ⁄ μ and s

Class exercise:

The Taj service station has five mechanics each of whom can service a scooter in one hour on an average. The scooters are registered at a single counter and then sent for servicing to different mechanics. Scooters arrive at the service station at an average rate of two scooter per hour. Assuming arrival follow Poisson and service time exponential, find

1. System Utilisation2. The probability that the system is idle3. The probability that there shall be three scooters in the

service center4. Expected number of scooters waiting in the queue