# Assumptions

The functions in the Queueing ToolPak fall into two categories.

## QTPMMS Functions

Functions in the first category have names that begin with "QTPMMS_" and are based on the following assumptions.

• Arrival Process: Arrivals are assumed to come from a stationary Poisson process. Two equivalent ways of saying this are (1) the number of arrivals in a fixed time interval is Poisson distributed, or (2) the times between arrivals are exponentially distributed.
• Waiting Room: Customers wait in a single queue (which may be spatially dispersed). It is possible to specify a finite capacity for the queue; otherwise the queue is assumed to have unlimited capacity.
• Service Process: Customers are served first-come-first-served by one of several parallel and identical servers. Times to complete service are exponentially distributed.

The functions may provide adequate approximations even when some of these assumptions do not hold exactly, but strictly speaking these assumptions are required for the functions to give valid results.

A queueing system that satisfies these assumptions is commonly referred to as an M/M/s system, where the first "M" signifies a "memoryless" arrival process, the second "M" signifies a memoryless service process, and "s" is the number of servers. If the queue has a capacity C, then the system is denoted M/M/s/s+C.

More detail about the formulas for an M/M/s/s+C system can be found in survey management science textbooks (see http://www.spreadsheetanalytics.com for one list of such books). The mathematical derivations can be found, for example, in Fundamentals of Queueing Theory, 3rd Edition, by Donald Gross and Carl Harris, John Wiley & Sons, 1997.

## QTPGGS Functions

Functions in the second category have names that begin with "QTPGGS_".  These functions are based on the following more general assumptions.

• Arrival Process: Times between arrivals ("interarrival times") are independent, but can follow any distribution.
• Waiting Room: Customers wait in a single queue (which may be spatially dispersed). The queue is assumed to have unlimited capacity.
• Service Process: Customers are served first-come-first-served by one of several parallel and identical servers. Times to complete service can follow any distribution.

A queueing system that satisfies these assumptions is referred to as a G/G/s system, where the "G"s signify that both the interarrival time distribution and the service time distribution are "General," i.e., they are not limited to an exponential distribution or any other specific distribution.

Exact results are difficult to compute for G/G/s systems, and require detailed knowledge of the distribution of interarrival times and the distribution of service times.  Instead, the functions in this category provide approximate values and require only the mean and standard deviation of the interarrival time distribution and the service time distribution.  The approximations used are described in the following articles:

1) Whitt, W. (1993) "Approximations for the GI/G/m Queue," Production and Operations Management, 2(2): 114-161.

Equation 2.24 is used for QTPGGS_Wq.  QTPGGS_W, QTPGGS_Lq, and QTPGGS_L are then determined from QTPGGS_W = QTPGGS_Wq + 1/(Service Rate), QTPGGS_Lq = (Arrival Rate) * QTPGGS_Wq, and QTPGGS_L = (Arrival Rate) * QTPGGS_W

2) Whitt, W. (2003) "A Diffusion Approximations for the G/GI/n/m Queue," Operations Research, forthcoming.

Equations 1.1, 4.8, and 4.12 are used for QTPGGS_PrWait.

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