Suppose i have a single web server machine on which clients send requests in form of poisson distribution and all the clients have different service times,now i am confused whether i will take this scenario as M/G/1 or M/M/1. I am also confused with 2nd letter M ie in M/M/1 2nd letter M stands for random what does it mean? In case of web server clients have different service time can i say "the clients have random service times? Plz explain
The standard notation of $A/B/C$ in queuing theory is as follows:
$A$ - The arrival process. $B$ - The service process. $C$ - The number of servers.
The most common assumption for both arrival and service is $M$ which stands for memoryless, i.e. exponential inter-arrival times (or exponential service times). $G$ Stands for a general unspecified distribution. Other common examples are $D$ for deterministic and $Er$ for Erlang.
So to answer your question, you need to specify if you assume exponential service times and then you have an $M/M/1$ system, or a general service distribution and then you have an $M/G/1$ system.
Note that it is very common to deal with $M/G/1$ because many of the steady-state results that appear in $M/M/1$ hold in the general case as well. This is not true if you change the arrival distribution ($G/M/1$ system). You can read up on the PASTA property if you are interested.