Sengupta (1989) showed that, for the first-come–first-served (FCFS)G/G/1 queue, the workload and attained waiting time of acustomer in service have the same stationary distribution. Sakasegawaand Wolff (1990) derived a sample path version of this result, showingthat the empirical distribution of the workload values over a busyperiod of a given sample path is identical to that of the attainedwaiting time values over the same period. For a given sample path of anFCFS G/G/s queue, we construct a dual sample path of adual queue which is FCFS G/G/s in reverse time. It isshown that the workload process on the original sample path isidentical to the total attained waiting time process on the dual samplepath. As an application of this duality relation, we show that, for atime-stationary FCFS M/M/s/k queue, the workload processis equal in distribution to the time-reversed total attained waitingtime process.
"A duality relation between the workload and attained waiting time in FCFS G/G/s queues." J. Appl. Probab. 50 (1) 300 - 307, March 2013. https://doi.org/10.1239/jap/1363784441