Waiting Line Theory Flashcards
Someone or something that is in
need of some type of service
Customer or element
a process or system that
performs the services to the
customer
Service facility
represents any type of attention
to satisfy customer or element
needs
Service
“The flow of customers from an infinite or finite population towards the service facility forms a queue or waiting line on account of lack of capability to serve all at a time.”
Service station
Number of customers waiting to be serviced.
Queue
A system having a service facility at which units of some kind (generically called “customers”) arrive for service
Queueing system
- Danish telephone engineer who did the original work on queueing theory.
- Effects on fluctuation demand on the utilization of automatic dialing equipment.
- By the end of World War II, waiting line models were extended to other kinds of problems.
A.K. Erlang, 1905
Four basic structures of waiting line theory
- Single Phase, Multiple Channel
- Single Phase, Single Channel
- Multiple Phase, Single Channel
- Multiple Phase, Multiple Channel
2 possible situations for a customer to acquire the service needed
- From customer to service center
- From service center to customer
When the customer is the one waiting…
- total number of customers > numher of facilities available
When the service center is the one waiting…
- total capacity of their system < total number of customers requiring service
Congestion
theoretical analysis of the waiting
line problem in telephone calls
A.K. Erlang, 1903
Developed the theory further
Mills and Thornton Fry, 1927
systematic and mathematical
approach to waiting line problem
David George Kendall, 1951
computerized reservation of
rail journey
1951 onwards
Queue Process
Arrival -> Waiting line -> Units served -> Leaving the queue
_____ __ ______ to the service system greatly depends on the nature of size of the population, which may be infinite or finite.
Size of arrivals
Arrival time distribution, customers arrive in Poisson or Completely random fashion.
Constant or Random
In queuing context, it refers to the space available for the arrivals to wait before being taken to service. The space available may be limited or unlimited.
Capacity of the Service System
The length of the queue or the waiting time of a customer or the idle time of the service facility mostly depends on the ________ _______
Customer Behavior
This behavior signifies that the
customer does not like to join the queue seeing the long length of it.
Balking
Customer joins the queue and after waiting for a certain time loses his patience and leaves the queue
Reneging
Customers may collaborate and only one of them may stand in the queue.
Collusion
If there are a number of waiting lines
depending on the number of service
stations.
Jockeying
Service Facility Design may be
Single channel or Multi channel
Queue service discipline may be
FIFO, LIFO, SIRO, Service by priority (which may be preemptive or non-preemptive)
probability distribution of queue length or the number of persons in the system at any point of time
Queue Length (Queueing Problem)
time spent by a customer in the queue before the commencement of his service
Waiting time (Queueing Problem)
- During the ____ _____, no customer is present in the system.
- During his _____ ____, some more customers will arrive and will be served in their turn according to the system discipline.
Idle Time, Busy Time
(Average Idle Time or Busy Time Distribution)
the rate of arrivals of customers is less than the rate of service and both are constant
Steady State
its operating characteristics or behavior are dependent on time
Transient State
Arrival rate of the system is larger than its service rate. Queue length will increase with time and theoretically it could build up to infinity.
Explosive State
(2) Common basic waiting line models
Poisson distribution and Negative Exponential distribution
A certain fast-food restaurant gets an
average of 3 visitors to the drivethrough per minute.
Poisson Distribution
Representative of the Poisson process, but describes the time between arrivals and specifies that these time intervals are completely random.
Negative Exponential Distribution
Corresponds to completely random arrivals and it is assumed that arrivals are completely independent of other arrivals as well as any condition of the waiting line.
Lamda (Arrivals per time unit)
Mean time between arrivals
1/y
the objective is to study the number of customers that enter the system, only arrivals are counted
Birth
Is a generalization of pure birth process, and has two types of state transitions: ‘_____’, which increases the current state variable by one, or
‘_____’, which decreases the current state variable by one
Birth-Death Process, Birth, Death
The reason for the common reference to rates in the discussion of arrivals and to times in the discussion of service is simply a _____ __ ________
Matter of practice
Commonly used symbols
muu and u
are single parameter distributions; that is, they are completely described by one parameter, the mean. For the ____ ____, the standard deviation is the square root of the mean,
and for the _____ _____ _____, the standard deviation is equal to the mean
Poisson distribution and Negative exponential distribution
It is to analyze and optimize systems
that involve waiting lines
Queueing Model
arrival and service rates are some unknown random variables.
Probabilistic Model
arrival and service rates are known and fixed
Deterministic Model
Either of the arrival and service rates is unknown random variable and other known and fixed.
Mixed Queueing Model
Average number of customer in system L or E (n)
L or E (n) = p / 1-p
Average number of time customer spends on the system W or E (v)
W or E (v) = L / y or
= 1 / u - y
Average number of customer in the queue E (L)
E (L) = p2 / (1 - p) or
= y2 / u (u - y)
Given:
Arrival (y)
Service (u)
Solution:
Service (u) = 1 / WS
(p) = y / u
Ave. # of Customer in system
L or E (n) = p / 1 - p
Ave. # of time customer spends
W or E (v) = L / y or 1 / (u - y)
Ave. # of customer in the queue
E (L) = p2 / (1 - p) or
y2 / u (u - y)
