Some queueing theory notations, however, have become so ubiquitous in the literature that there is no point being different, just for the sake of being different. Therefore, I do conform to using the conventional queueing symbols:
- λ for arrival rate
- μ for service rate
- ρ for server utilization
Sadly, I always have to confess to students that I have never read a reason why these symbols are used. I still haven't, but I did have a flash of inspiration that might explain them. I haven't done any deep historical research, so caveat lector.
It seems reasonable to suspect that the above conventions come from Agner Erlang, the Dane who developed and solved the first queueing model (M/M/m) in 1917. (BTW, I also use the standard Kendall notation.) Erlang was doing capacity planning for the Internet of his day; the telephone network. Back then, to make calls outside Copenhagen, you needed a "trunk line", which had to be reserved ahead of time. The Copenhagen telephone company wanted to know how much trunk-line capacity they needed (measured today in Erlangs) to handle the expected number of calls. Some things never change.
I have a copy of Erlang's 1917 paper, so I looked it up. However, he does NOT use the modern Greek symbols. Instead, he uses x and y in the following way:
- x: number of trunk lines
- y: average number of calls/unit time or "traffic intensity" (Erlang's term)
- a: traffic intensity per line such that, a = y/x or ax = y
In modern Greek notation, this becomes:
- m: number of trunk lines
- λ S: total traffic intensity (where the mean service period S = 1/μ)
- ρ: traffic intensity per line such that, m ρ = λ S (Little's law)
So, it looks to me like the modern notation we use today comes from a later time, and perhaps from the probability theorists (such as Markov and Kolmogorov) who got hold of this stuff during the 1930's or thereabouts. The origins probably derive from something like the following:
- λ: Greek 'L' for the statistical mean of the exponential distribution for upward Markov transitions or interarrival periods
- μ: Next letter 'M' in the Greek alphabet for exponential downward Markov transitions or service periods
- ρ: Greek 'R' for "ratio": ρ = λ / μ (Little's law)
Just a guess on my part, but it seems plausible. If you happen to know the accurate historcial background, submit a comment.