Operations Research Models and Methods / Computation

Queuing Add-in


User Functions

 

The steady state results for Poisson queueing systems are computed by user defined functions provided by the Queueing Add-in. The function names all have the Q_ prefix. They are listed below with their parameters. The values are from the Q_Sample example. Most of the functions have a single argument which is the range defining the queue. The range is defined by the name given the queue. For example the first function would be Q_type(Q_Sample).

The range defined by a queue name holds five parameters: Lam = arrival rate when the system is empty, Mu = service rate for each server, ss = number of servers, SS_Max = maximum number in the system, PPop_Max = Size of the calling population. Leaving a blank for either of the latter two parameters or using non-numerical values implies that the parameter is infinity. For the Q_Sample example the parameters are in the range B2:B6 and take the values:

(5, 3, 2, ***, ***)

The *** used for the last two parameters indicate that the maximum number in the system and the population size are both infinite.


Function

Notation

Result

Q_type(Q_Sample):
Determines the type of queue using Kendall's notation.

Type =

M/M/3

Q_L(Q_Sample):
Computes the mean number in the system.

L=

6.011236

Q_W(Q_Sample):
Computes the mean number in the system.

W =

1.2022472

Q_Lq(Q_Sample):
Computes the mean number in the queue.

Lq=

3.511236

Q_Wq(Q_Sample):
Computes the mean time in the queue.

Wq =

0.7022472

Q_Ls(Q_Sample):
Computes the mean number in service.

Ls=

2.5

Q_Ws(Q_Sample):
Computes the mean time in service.

Ws =

0.5

Q_LamB(Q_Sample):
Computes the throughput of the station.

LamB =

5

Q_Eff(Q_Sample):
Computes the efficiency of the servers.

Eff =

0.8333333

Q_P0(Q_Sample):
Computes the probability of 0 in the system.

P0 =

0.0449438

Q_PB(Q_Sample):
Computes the probability that all servers are busy.

PB =

0.7022472

Q_PF(Q_Sample):
Computes the probability that the system is full.

PF =

0

Q_FNext(k, Q_Sample):
The FNext function computes the factor to obtain the next probability in a series of state probabilities. The function must be multiplied by the previous probability. k is the index of the state computed.
P(1) = P(0)*FNext(1, Queue)

P(1) =

0.1123596

Q_Pn(k, Q_Sample):
Computes the probability of n customers in the system. Illustrated for 11.

P(11) =

0.02722

Q_PTq(time, Q_Sample):Computes the cumulative probability distribution of the waiting time in the queue. An example of this function is shown below.

PTq(0.5) =
0.4259344
Non-Markovian Queues

Operations Research Models and Methods
by Paul A. Jensen and Jon Bard, University of Texas, Copyright by the Authors