Waiting Line Management

Where the Time Goes?

In a life time, the average person will spend:

SIX MONTHS Waiting at stoplights

EIGHT MONTHS Opening junk mail

ONE YEAR Looking for misplaced 0bjects TWO YEARS Reading E-mail FOUR YEARS Doing housework FIVE YEARS Waiting in line SIX YEARS Eating

12-2

Waiting Line ……..

Understanding waiting lines or queues and learning how to manage them is one of the most important areas in operations management. It is basic to creating schedules, job design, inventory levels, and so on. In our service economy we wait in line every day, from driving to work to checking out at the supermarket. We also encounter waiting lines at factories— jobs wait in lines to be worked on at different machines, and machines themselves wait their turn to be overhauled. In short, waiting lines are pervasive.

Waiting lines are non-value added occurrences

Waiting Lines

Often called queuing theory Waiting lines are common situations Useful in both

manufacturing and service areas

Service Capacity vs Waiting Line Trade Off

Goal of queuing analysis is to minimize the sum of two costs

Customer waiting costs

Service capacity costs

The Queuing SystemThe queuing system consists essentially of three major components: (1) thesource population and the way customers arrive at the system, (2) the servicing system, and (3) the condition of the customers exiting the system (back to source population or not?).

The Queuing System :Factors

• Length• Number of Lines• Queue Discipline

Line Structures• Single channel, single phase : This is the simplest type of waiting line

structure.• Single channel, multiphase : A car wash is an illustration because a

series of services (vacuuming, wetting, washing, rinsing, drying, window cleaning, and parking) is performed in a fairly uniform sequence.

• Multi channel, single phase : Tellers’ windows in a bank and checkout counters in high-volume department stores exemplify this type of structure.

• Multi channel, multiphase : This case is similar to the preceding one except that two or more services are performed in sequence.

• Mixed :Under this general heading we consider two subcategories: (1) multiple-to single channel structures and (2) alternative path structures.

Line Structures

Queuing System Designs

Departuresafter service

Single-channel, single-phase system

Queue

Arrivals

Single-channel, multiphase system

Arrivals Departuresafter service

Phase 1 service facility

Phase 2 service facility

Service facility

Queue

A family dentist’s office

A McDonald’s dual window drive-through

Queuing System Designs

Multi-channel, single-phase system

Arrivals

Queue

Most bank and post office service windows

Departuresafter service

Service facility

Channel 1

Service facility

Channel 2

Service facility

Channel 3

Queuing System Designs

Multi-channel, multiphase system

Arrivals

Queue

Some college registrations

Departuresafter service

Phase 2 service facility

Channel 1

Phase 2 service facility

Channel 2

Phase 1 service facility

Channel 1

Phase 1 service facility

Channel 2

Properties of Some Specific Waiting Line Models

Probability DistributionsThe sources of variation in waiting-line problems come from the random arrivals of customers and the variations in service times. Each of these sources can be described with a probability distribution.

Arrival DistributionCustomers arrive at service facilities randomly. The variability of customer arrivals often can be described by a Poisson distribution, which specifies the probability that n customers will arrive in T time periods :

for n = 0,1,2….. whereP = probability of n arrivals in T time periodsλ = average number of customer arrivals per periode = 2.7183.The mean of the Poisson distribution is λT, and the variance also is λT. The Poisson distribution is a discrete distribution, that is, the probabilities are for a specific number of arrivals per unit of time.

Probability DistributionsService Time DistributionThe exponential distribution describes the probability that the service time of the customer at a particular facility will be no more than T time periods. The probability can be calculated by using the formula :

Where , μ = average number of customers completing service time period t = service time of the customer T = target service time

Single Server ModelThe simplest waiting line model involves a single server and a single line of customers. To further specify the model, we make the following assumptions :1. The customer population is infinite and all customers are patient.2. The customers arrive according to a Poisson distribution, with a mean arrival rate of λ3. The service distribution is exponential, with a mean service rate of μ.4. The mean service rate exceeds the mean arrival rate.5. Customers are served on a first-come, first-served basis6. The length of the waiting line is unlimited.

With these assumptions, we can apply various formulas to describe the operating characteristics of the system :

Multiple Server ModelWith the multiple-server model, customers form a single line and choose one of s servers when one is available. The service system has only one phase. We make the followng assumptions in addition to those for single-server model: There are s identical servers, and the service distributon for each server is exponential, with a mean service time of 1/μ. It should always be the case that sμ exceeds λ.With these assumptions, we can apply several formulas to describe the operating characteristics of the service system :

Little’s Law

One of the most practical and fundamental laws in waiting line theory is Little’s Law, which relates the number of customers in a waiting line system to the arrival rate and the waiting time of customers. Using the same notation we used for the single server and multiple server models, Little’s Law can be expressed as L=λW or Lq=λWq. This relationship holds for a wide variety of arrival processes, service time distributions, and number of servers. The practical advantage of Little’s Law is that you only need to know two of the parameters to estimate the third.

Finite Source ModelWe now consider a situation in which all but one of the assumptions of the single server model are approximate. In this case, the customer population is finite, having only N potential customers. If N is greater than 30 customers, the single server model with the assumption of an infinite customer population is adequate. Otherwise, the finite source model is the one to use. The formulas used to calculate the operating characteristics of this server system include the following :

Measuring Queue Performance

1. Average time that each customer or object spends in the queue2. Average queue length3. Average time each customer spends in the system4. Average number of customers in the system5. Probability that the service facility will be idle6. Utilization factor for the system7. Probability of a specific number of customers in the system