How hard is it to compute the distribution of the number-in-system, 
etc. for various types of queue?

For M/M/1, it's easy; for G/G/c, it's basically impossible.
Here we give ratings for how hard it is to compute (on a computer) _exact_ performance measurements.
Note that there are pretty good _approximations_ for most queueing systems, at least for
their average performance (e.g. the average number in system).

M = Markovian, which means Memoryless or exponential or a Poisson arrival process.
PH = PHase-type (combinations of exponentials)
D = Deterministic
G = General (includes all of the above and a lot more)

1=easy, 5=hard
Continuous Time queues

1-server:
        Svc
        M       PH      D       G
Arr
M       1       2or3    2       2
PH      3       2.5     2.5     2.5
D       1.5     4       1       5
G       2       2.5     5       5

multi-server:

        Svc
        M       PH      D       G
Arr
M       1       4       3       5
PH      3       4       4.5     5
D       3       4       1       5
G       3       4       5       5

Rough system for choosing ratings:
1       closed-form
2       Laplace transforms
3       intricate but not huge state space
(maybe 2 and 3 should be tied?)
4       huge state space
5       intractable


By the way, M/M/c/0 is the Erlang-B model,
M/M/c is the Erlang-C model,
M/M/c + abandonments is what some call the Erlang-A model.