Reasonable/unreasonable exponentially distributed interarrival (service) times

Question

As the title suggests, when do we consider the exponentially distributed interarrival (service) times to be reasonable or unreasonable for a queuing system?

I'm pretty clueless about this (or maybe I just can't see the obvious answer) as I'm new to the topic of queuing theory..

Answer 1

Exponentially-distributed inter-arrival times means that the arrivals are a Poisson process. There are three axioms that characterize a Poisson process. There are various formulations, for example:

1. The probability of two arrivals at the same instant is 0.

2. The number of arrivals in two non-overlapping time intervals is independent.

3. The probability of an arrival occurring in a short time interval is proportional to the length of the interval.

So, it come down to the question of whether the arrival process satisfies these axioms, at least approximately. Note that this may be true at some times of day, but not others, or it may be true with different arrival rates in different periods. You should Google "axioms for Poisson process" to see other formulations.

Similar remarks hold for exponential service times. The distribution of the time to completion doesn't depend on how long the service has taken so far. For example if we only have two kinds of jobs, and one takes a short time and the other a long time, then exponential service times doesn't seem like a good fit, because if the service time is already twice the length of a short job, the probability that it's a long job has surely increased.

I'm no queueing theorist, although I took a few courses a long time ago. It always seemed to me that the assumption of exponential service time was mainly motivated by the fact that it greatly simplifies the math. (Of course, there may be good reasons which were simply not discussed in my classes, which focussed on the math. In a particular case, I'd like to get a list of actual service times, and put them on a histogram.)

Answer 2

Here are a few "real-world" considerations that might cause you to reject exponential interarrival or service times.

• Customers can be broken into classes with distinctly different arrival rates or distinctly different service rates. Example: the bottle crusher at my local supermarket. Customer group #1 is people like me recycling a six-pack or a few cans. Customer group #2 is people who just threw big parties, or people who scavenge throw-aways for the deposits, who come in with shopping carts full of cans/bottles.
• Either arrivals or services follow a seasonal pattern. The IATs or STs might be "locally exponential", but not over the long haul. If you need examples, go to any mall just before Christmas.
• Events or interventions mess with the arrival or service rates. Stores having advertised sales come to mind.
• Arrivals stimulate (or inhibit) arrivals. Stimulation might occur if, for example, people spot a sale and immediately spread the word via social media, or if a crowd draws the attention of passers-by. Inhibition might occur if large numbers of arriving customers result in a parking lot filling up, preventing subsequent arrivals (which could be viewed as a capacitated queue) or in traffic congestion that scares off customers.
• Arrivals occur in clumps. For example, a store (or mall) might get some customers who arrive more or less singly by car or bike but others who come by bus (arriving en masse when the bus gets there). Restaurants and shops in ports where cruise ships dock would be another example.
• Queues expedite or inhibit service. In some situations, tellers or cashiers might work faster when they see a line. In others (I'm thinking government offices), they might work slower when they see a line.