The steady state results for Poisson Queuing systems are computed by user defined functions provided by the Queuing Add-in. These functions are in the Excel function list under the heading User Defined Functions. 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, 2, 3, ***, ***)

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

Operations Research Models and Methods
Internet
by Paul A. Jensen