48 terms

Kendall Notation:

- Developed to allow the key characteristics of a specific queuing model to be described in an efficient manner.

- Simple queuing models can be described by there parameters in the following general format:

1 / 2 / 3

- Simple queuing models can be described by there parameters in the following general format:

1 / 2 / 3

1 / 2 / 3

Parameter 1 identifies?

Parameter 1 identifies?

Identifies the nature of the arrival process.

M = Markovian interarrival times

(Following an exponential distribution)

D = Deterministic interarrival times (Not Random)

M = Markovian interarrival times

(Following an exponential distribution)

D = Deterministic interarrival times (Not Random)

1 / 2 / 3

Parameter 2 identifies?

Parameter 2 identifies?

Identifies the nature of the service times.

M = Markovian interarrival times

(Following an exponential distribution)

G = General service times (following a nonexponential distribution)

D = Deterministic interarrival times (Not Random)

M = Markovian interarrival times

(Following an exponential distribution)

G = General service times (following a nonexponential distribution)

D = Deterministic interarrival times (Not Random)

1 / 2 / 3

Parameter 3 identifies?

Parameter 3 identifies?

Indicates the number of servers available.

M / M / 1 queue refers to a queuing model in which :

- The time between arrivals follows an exponential distribution

- The service times follow an exponential distribution

- There is one server

- The service times follow an exponential distribution

- There is one server

M / G / 3 queue refers to:

A model in which the interarrival times are assumed to be exponential, the service times follow some general distribution, & three servers are present.

Queuing models are used to:

- describe the behavior of queuing systems

- determine the level of service to provide

- evaluate alternate configurations for providing service

- determine the level of service to provide

- evaluate alternate configurations for providing service

T or F : Random service times from an exponential distribution can assume any positive value.

TRUE

Queue :

- a waiting line

Queuing Theory :

- body of knowledge dealing with waiting lines

M / M / s model is appropriate for analyzing queuing problems when these specific assumptions are met:

Assumptions:

- There are s servers, where s is a positive integer

- Arrivals follow a Poisson distribution and occur at an average rate of l per time period.

- Each server provides service at an average rate of u per time period, and actual service times follow an exponential distribution.

- Arrivals wait in a single FIFO queue and are serviced by the first available server.

- l < s u

- There are s servers, where s is a positive integer

- Arrivals follow a Poisson distribution and occur at an average rate of l per time period.

- Each server provides service at an average rate of u per time period, and actual service times follow an exponential distribution.

- Arrivals wait in a single FIFO queue and are serviced by the first available server.

- l < s u

Results for the M / M /s models assume that:

The size or capacity of the waiting area is infinite, so that all arrivals to the system join the queue and wait for service.

What does the final assumption indicate?

l (symbol for arrival rate) < s u

l (symbol for arrival rate) < s u

That the total service capacity of the system, s u , must be strictly greater than the rate at which arrivals occur l (symbol for arrival rate).

If the arrival rate exceeds the system's total service capacity, then :

The system would fill up over time, and the queue would become infinitely long.

** queue will also become infinitely long even if the average arrival rate is equal to the average service rate s u

** queue will also become infinitely long even if the average arrival rate is equal to the average service rate s u

Will there be times when the servers are idle ?!

- Yes, and this idle time is lost forever.

- The servers will not be able to make up for this at other times when the demand for service is heavy.

- The servers will not be able to make up for this at other times when the demand for service is heavy.

Infinite Queue :

Queue which continues to expand

(calling units are coming faster than server can handle them).

(calling units are coming faster than server can handle them).

Finite Queue Length :

The size or capacity of the waiting area has a restriction

In some problems, the amount of waiting area is limited.

This means that rather than wait for service, units will balk.

In some problems, the amount of waiting area is limited.

This means that rather than wait for service, units will balk.

Balk :

Refers to an arrival that does not join the queue because the queue is full or too long.

M / M / s model with finite population :

* these queuing models have a finite arrival (or calling) population

* the average arrival rate for the system changes depending on the number of customers in the queue

* the average arrival rate for the system changes depending on the number of customers in the queue

M / M / s model with finite population is appropriate for analyzing queuing problems where the following assumptions are met:

- there are s servers, where s is a positive integer

- there are N potential customers in the arrival population

- the arrival pattern of each customer follows a Poisson distribution with a mean arrival rate of l per time period

- each server provides service at an average rate of u per time period, and actual service times follow an exponential distribution

- arrivals wait in a single FIFO queue and are serviced by the first available server

** note the avg arrival rate for this model is defined in terms of the rate at which each customer arrives

- there are N potential customers in the arrival population

- the arrival pattern of each customer follows a Poisson distribution with a mean arrival rate of l per time period

- each server provides service at an average rate of u per time period, and actual service times follow an exponential distribution

- arrivals wait in a single FIFO queue and are serviced by the first available server

** note the avg arrival rate for this model is defined in terms of the rate at which each customer arrives

M / G / 1 Queuing Model :

- Enables us to analyze queuing problems in which service times cannot be modeled accurately using an exponential distribution

- this queuing model is remarkable because it can be used to compute the operating characteristics for any one-server queuing system where arrivals follow a Poisson distribution and the mean u and standard deviation o of the service time are known.

^^ essentially can be used when service times are random with known mean and standard deviation

- this queuing model is remarkable because it can be used to compute the operating characteristics for any one-server queuing system where arrivals follow a Poisson distribution and the mean u and standard deviation o of the service time are known.

^^ essentially can be used when service times are random with known mean and standard deviation

Renege :

- leave a queue before being served

Jockey :

- switch from one queue to another

Typical operating characteristics of interest include:

U

U

Utilization factor, % of time that all servers are busy

P(0)

- Probability that there are no zero units in the system

Lq

- Average number of units in line waiting for service

L

- Average number of units in the system (in line & being served)

Wq

- Average time a unit spends in line waiting for service

W

- Average time a unit spends in the system (in line & being served)

Pw

- Probability that an arriving unit has to wait for service

Pn

- Probability of n units in the system

Arrival Rate :

- the manner in which customers arrive at the system for service

- model the arrival process in a queuing system as a Poisson random variable

- model the arrival process in a queuing system as a Poisson random variable

Interarrival Time:

- random amount of time that is likely to transpire between arriving calls

- follows an exponential probability distribution with mean 1 / l

^^ if the # of arrivals in a given period follows a Poisson distribution with mean l

- follows an exponential probability distribution with mean 1 / l

^^ if the # of arrivals in a given period follows a Poisson distribution with mean l

Exponential Distribution :

- plays a key role in queuing models

- one of the few probability distributions that exhibits the memory-less property

- one of the few probability distributions that exhibits the memory-less property

Markov:

First to recognize the memory less property of certain random variables; thus memoryless property is referred to as the Markov or Markovian property

Queue Time:

Amount of time a customer spends waiting in line for service to being

Service Time:

Amount of time a customer spends at a service facility after the actual performance of service begins

Queue time VS Service time

SERVICE TIME DOES NOT INCLUDE QUEUE TIME!

Service Rate :

- denoted by u

- representing the average number of customers (or jobs) that can be served per time period

- average service time per customer is 1 / u time periods

- variance of the service time per customer is (1/u)^2 time periods

- representing the average number of customers (or jobs) that can be served per time period

- average service time per customer is 1 / u time periods

- variance of the service time per customer is (1/u)^2 time periods

What Operating Characteristic will give you the "Probability that there are no trucks in the system" ?

P(0)

What Operating Characteristic will give you the "average number of trucks waiting for service"?

Lq

What Operating Characteristic will give you the "average time a truck waits for the loading/unloading service to begin"?

Wq

What Operating Characteristic will give you the "probability a new arrival will have to wait"?

- Pw

- Probability that a customer waits.

- Probability that a customer waits.

L - Lq =

- Average # of number of units actually being serviced

W - Wq =

- Average time it takes a unit to be serviced

With respect to farther improving profit, would you make any other recommendations to The Three Little Pigs regarding the operation of their checkout counters?

I would recommend that Hansel and Gretel find something for their employees to do when they are not busy operating their checkout stand.

If all the operating characteristics of a M / M / S = #DIV/0 then what does this mean for the queue?

This means the queue of customer/trucks/units of whatever would increase endlessly.

If the Lq, L, W, & Wq of a M/M/S = a negative number then what does this mean for the queue?

It means we have an infinite queue.