Queueing Theory

STUDY
PLAY
queue
waiting line
objective of designing a "good" queueing system
to help an organization perform optimally according to some criterion (wait time, service level, line length, profit, probabilities of wait time)
3 components of a queueing system
1. arrivals
2. waiting in line
3. service facility
trade-offs of designing a queueing system
balancing customer service and profitability
deterministic arrival process
customers arrive according to some known schedule
stochastic arrival process
customers arrive individually and randomly, often modeled as a Poisson distribution
3 conditions of a Poisson distribution
1. orderliness
2. stationarity
3. independence
Poisson: orderliness
during any time interval, at most one customer will arrive at the service facility
Poisson: stationarity
for a given time frame, the probability of customer arrivals remains the same for each incremental time interval
Poisson: independence
the arrival of one customer has no influence on the arrival of another customer
queueing notation
arrival process / service process / number of servers / max customers in the system / max customers in the population
M/M/1
single server, single channel, poisson arrival distribution, exponential distribution for service
M/M/k
1. customers arrive according to a Poisson process
2. Service times follow an exponential distribution
3. Each of the k servers works at an average rate of u
the manner in which units receive their service, such as FCFS is the _______.
queue discipline
the assumption of exponentially distributed service times indicates that ______.
approximately 63% of the service times are less than the mean service time
For a multiple server queueing system, as assumed _____.
each server had the same service rate.
M/G/1
service time takes a general form--need to know the mean and standard deviation of the service times.
what are the two special cases of a M/G/1?
M/D/1- deterministic service times, customer served at a constant rate

M/En/1- erlangian distribution for service times
M/M/k/F
Poisson arrival rate with a mean of lambda

k servers, each with exponential service time distribution with a mean rate of mu

Upper limit of F customers who can present in the system at any one time
Po
probability there are no customers in the system
Pn
probability there are n customers in the system
L
average number of customers in the system
Lq
average number of customers in the queue
W
average time a customer spends in the system
Wq
average time a customer spends in the queue
Pw
probability that an arriving customer must wait for service
,o
utilization rate of each server (percentage of time that each server is busy)
for many waiting line situations, the arrivals occur randomly and independently of other arrivals and it has been found that a good description of the arrival pattern is provided by _________.
poisson probability distibution
the equations provided in the textbook for computing operating characteristics apply to a waiting line operating _________.
a steady-state
the machine repair problem is an application of the M/M/1 model with ____________.
a finite calling population
Memoryless Property (Markovian)
1. exponential distribution has the special property making it memoryless distribution