Consider a G$/$G$/1$ queue with the FIFO service discipline, interarrival times

u$1,$ u$2,...,$ and service times v$1,$ v$2,....$ Assume that the system is initially empty. For i $>=1,$ let Si denote the system time $($also called sojourn time$)$ of the ith arriving customer $($i$.$e$.,$ Si $=$ Wi $+$ vi for all i$,$ where Wi denotes the time the ith arriving customer waits in queue prior to entering service$).$ Express Si$+1$ as a function of Si$,$ ui$+1,$ and vi$+1.$ Explain $($your answer should resemble Lindley$$s equation, but apply to the system times S$1,$ S$2,...,$ rather than the waiting times W$1,$ W$2,...)$