# Mean service time of a $M/E_2/1$ queueing system?

Consider a $M/E_2/1$ queueing system, where the customer arrival rate is $\lambda$ and the service time distribution has a gamma distribution with parameters $2$ and $\mu$, i.e. with p.d.f. $\mu^2te^{-\mu t}$ , $t ≥ 0$

(1) How can I determine the mean of the service time distribution?

(2) What is the traffic intensity $\rho$ of the system in terms of the parameters $\lambda$ and $\mu$?

My reasoning thus far: wouldn't I simply take the mean of a Gamma(2,$\mu$) distribution and thus just say the mean service time is $\frac{2}{\mu}$ but I'm guessing there has to be something more to it than that?

As for the triffic intensity $\rho$ I do not know what it would be for an erlang-2 queueing system? My thinking is (using $c=1$ for the number of servers):

$\rho = \frac{"mean.service.time"}{c*"mean.customer.interarrival.time"} = \frac{"mean.service.time"}{c*\frac{1}{"arrival.rate"}} =\frac{\frac{2}{\mu}}{1*\frac{1}{\lambda}}=\frac{2\lambda}{\mu}$

Note: Follow up question: Mean length of time spent queueing in $M/E_2/1$ system?

• I don't think there's anything more to working out the mean service time than what you did. – Glen_b -Reinstate Monica Jan 26 '14 at 2:59
• @Glen_b Maybe I was over complicating it in my head. However, I don't know what the traffic intensity $\rho$ would be? – Clair Crossupton Jan 26 '14 at 10:56
• Sorry, it's more decades than I'd like to mention publicly since I did any queueing theory -- so I don't remember what the definition of traffic intensity is, and google wasn't much help. If you could define it, perhaps, or point to one, that might help. (Incidentally, questions such as these would normally carry the self-study tag (q.v.); do you need all five of those tags? I am thinking you could manage without one of the -process tags) – Glen_b -Reinstate Monica Jan 26 '14 at 22:04

1) Yes, the mean of the service time distribution is just the mean of the Gamma(2,$\lambda)$ distribution.

2) The traffic intensity of the system is the arrival rate / the service rate, in this case:

$$\rho = \lambda \mu / 2$$.

These questions are a bit simplistic, but perhaps the intent of the first is to get you away from thinking of the mean service time as denoted by $\mu$ in all cases.

Interestingly enough, the Gamma distribution with integer shape parameter is also known as the "Erlang" distribution (hence the $E_2$ in the descriptor of the queuing system), and in telecommunications the various measures of traffic intensity (which are different than the queuing theory definition) are in "erlangs".

• Hiya, I got a different a different traffic intensity $\rho$ to what you have? – Clair Crossupton Jan 30 '14 at 10:05
• What do you have for traffic intensity? – jbowman Jan 30 '14 at 14:21
• I derived it as $\rho=\frac{2\lambda}{\mu}$ (see bottom of my question). – Clair Crossupton Jan 30 '14 at 15:51
• @jbowman your formula for $\rho$ looks consistent with the OP's (aside from the inclusion of the factor $c=1$), but (unless I made an error) it looks to me like you put in the service time for the service rate. – Glen_b -Reinstate Monica Jan 30 '14 at 22:17