$1.$ Consider an M$/$M$/1/2$ system with input rate $\backslash$lambda $.$ When a customer arrives to an empty system, the server proceeds at rate $\backslash$mu $.$ However, if a second customer arrives while the first is still in service, the server speeds up to a rate $2\backslash$mu $,$ and continues at rate $2\backslash$mu until the system empties again. Only one customer will be in service at a time. Note that the system has limited storage and can hold at most one in service and one in queue. Find P$0=$ P$[$systemisempty$].$ Be sure to express your answer explicitly in terms of $\backslash$lambda $,$ and $\backslash$mu only. $2.$ In a TA$-$office there are half as many workstations as there are TA$$s$.$ Assume that the amount of time each TA spends away from the office is an exponentially distributed random variable with mean $1/\backslash$lambda $.$ Also assume that each time the TA returns to the office s$/$he needs to use a workstation for an exponentially distributed amount of time with mean $1/\backslash$mu $.($Also assume the TA will wait for and use the workstation and then leave the office on each visit.$)$ As a TA$,$ you would prefer the office in which the probability of finding an idle workstation in the office is highest. Based on these assumptions: Find the probability that a workstation is idle in a two$-$TA office, in terms of $\backslash$lambda and $\backslash$mu $.$ Find the probability that a workstation is idle in a four$-$TA office in terms of $\backslash$lambda and $\backslash$mu $.$ For $\backslash$lambda $=\backslash$mu $,$ which system has the lower expected waiting time for workstation, the two$-$TA or the four$-$TA office? If the two workstations in the four$-$TA office are replaced by a workstation that is twice as fast $($so that a TA only needs it for $1/(2\backslash$mu $)$ on average$),$ what is the probability of finding it idle?