# Kendall notation's “General distribution”, what does that mean?

The first and second parameters for the Kendall's notation may have a G value, which stands for General distribution, see here.

But what does that mean? What is a general distribution?

Queueing theory uses Kendall's notation, as you described. There are three components describing the behavior of a queue:

1. The customers arriving for service, which is usually described by a Poisson process (random arrivals), but sometimes by non-Poisson processes or even deterministic arrivals rates
2. The time required to service each customer, which is usually described by a probability distribution, e.g. exponential or gamma (Erlang) distributed service times, possibly deterministic though.
3. The number of service providers, a positive integer value.

Generally general case

In the most general case, the behavior of a queue would be described as G/G/c where G is an unknown rate of customer arrivals, with an unknown service time distribution, G, (which is NOT necessarily the same as the process that characterizes arrivals), and c is an integer value greater than or equal to one.

In such general terms, it doesn't make much sense! It is more accessible to start with a specific queue behavior, for example, the performance of a fixed hard disk-drive (i.e. DASD, or direct access storage device).

M/G/1

DASD performance is modeled most accurately as an M/G/1 queue. M means that customers, or requests for disk access, behave according to a Poisson process. This is referred to as a stochastic, or Markov process, thus the use of "M". The rate at which the disk drive is able to meet these requests for service is unknown. Since job service times can have an arbitrary distribution, this is designated by "G" for "general". Finally, if there is only one disk-drive, c = 1.

M/M/c

Let's consider another example, where customers arrive randomly (according to a Poisson process), with exponentially distributed service times. There are multiple servers. This would be described as an M/M/c queue.

This is the typical situation at Walmart, during the night shift (with few cashiers on duty), or at a bank with tellers, or when making a phone call for customer support. Customers arrive randomly (M). The time required to check out their groceries or answer their question is also random (M) e.g. when grocery queues don't have a "10 items or less" configuration for some checkers. Meanwhile, there are a fixed number of cashiers or telephone support staff on duty, we'll say five. This would be an M/M/5 queue.

• So G is just a "placeholder" for another distribution, right? – Marco A. Apr 29 '12 at 17:03
• Sometimes. I guess I would describe G as the general case where you just don't know what sort of service time distribution to expect. Unknown isn't the same as random, of course (that confused me when I first learned about this). Also, there are methods for characterizing a G distribution, because you might not EVER be able to determine what distribution would take the place of G. – Ellie Kesselman Apr 30 '12 at 20:17
• oh I thought that this stuff wasn't even used in real life jobs.. I thought it was merely theoretical, but seems that I'm wrong! I'm okay with the G general theory since I'm not required to study it for now (I'm following an academic course), I just wanted to understand what the G meant and you helped me in that. Do you have any experience with multi-class queues too? – Marco A. May 1 '12 at 9:53