For an M$/$M$/1$ queue with mean interarrival time $1\mathrm{.}25$ min. and mean service time $1$ minute, find all five of Wq$,$ W$,$ L$,$ Lq$,$ and $.$ For each interpret in words. Be sure to state all of the relevant units $($always$)$ and the time frame of operation.

Repeat problem $1,$ except assume that service times are not exponentially distributed, but rather $($continuously$)$ uniformly distributed between a $=0\mathrm{.}1$ and b $=1\mathrm{.}9.$ Note that the expected value of this uniform distribution is $($a$+$b$)/2=1,$ the same as the expected service time in Problem $1.$ Compare all five of your numerical results to those from problem $1$ and explain any differences intuitively with respect to the change in the distributions even though the expected values are the same. Hint: the standard deviation of the continuous uniform distribution between a and b is $(($b$-$a$)^2/12).$

Repeat problem $1,$ except assume that the service times are triangularly distributed between a $=0\mathrm{.}1$ and b $=1\mathrm{.}9,$ with mode m $=1\mathrm{.}0.$ Compare all five of your results with Problems $1$ and $2.$ Hint: the expected value of a triangular distribution between a and b with mode m$,$ such that a $<$ m $<$ b $,$ is $($a $+$m $+$b$)/3$ and the standard deviation is $(($a$^2+$ m$^2+$ b$^2-$am$-$ab$-$bm$)/18.)$

For each of problems $1,2,$ and $3,$ suppose that we$$d like to see what would happen if the arrival rate would increase by small steps. Create a spreadsheet or computer program and re$-$evaluate Wq$,$ W$,$ L$,$ Lq$,$ and $$ while increasing the arrival rate by $5\%$ over the original value until you reach a $100\%$ increase $($i$.$e$.,$ doubling the original arrival rate$).$ Plot each of the five values as a function of the increasing arrival rate.

Graduate Question Find all five of the steady state queueing metrics for an M$/$D$/1$ queue where $$D$$ denotes a deterministic distribution, i$.$e$.,$ the associated R$.$V$.$ is a non$-$varying constant, also called a $$degenerate$$ distribution. State parameter conditions for $$ and $$as done in the chapter. Arrival Rate: $\backslash$lambda

Service Rate: $\backslash$mu

Utilization Rate: $\backslash$rho $=\backslash$lambda $/\backslash$mu

Using the Pollaczek$-$Khinchine formula

Steady$-$State Average Time in the Queue: W$\_$q$=$

note Wq is $(1)/(2)$ the value of Wq for M$/$M$/1$

Steady$-$State Average Time in the System: W$=$

Steady$-$State Average Number in the Queue: L$\_$q$=$

Steady$-$State Average Number in the System: L$=$