31 terms

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

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

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

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/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

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