Rest Service Time of a M/M/1-FIFO queue

Question

How to derive the value for rest service time from arrival rate λ and mean service rate μ in a M/M/1-FIFO queue?

Is this correct? System time = Waiting time + η(service time) + (1-η)(rest service time)

edited: I am not clear on what rest service time is. Actually it is another thing that I wanted to know. Does below make sense?

Since,

System Time = Waiting Time + Service Time

and maybe,

Service Time = Time Served by Server + Other(Rest) Service Time

Answer 1

correct equation is, [Mean waiting time] = [service rate].[Mean number in queue] + [rest service time of a job in service]

[service rate].[Mean number in queue] <- actual time need to complete for all items in the waiting queue.

[Mean number in queue] <- [arrival rate λ].[Mean waiting time]; by Little’s law;