On ARL-unbiased charts to monitor the traffic intensity of a single server queue

Frontiers in Statistical Quality Control (Vol. 12), (2018), 87-112
https://doi.org/10.1007/978-3-319-75295-2_5

We know too well that the effective operation of a queueing system re- quires maintaining the traffic intensity rho at a target value rho0.
This important measure of congestion can be monitored by using control charts, such as the one found in the seminal work by Bhat and Rao (1972) or more recently in Chen and Zhou (2015).
For all intents and purposes, this paper focus on three control statistics chosen by Morais and Pacheco (2015a) for their simplicity, recursive and Markovian character:
- Xn, the number of customers left behind in the M/G/1 system by the nth depart- ing customer;
- Xˆn, the number of customers seen in the GI/M/1 system by the nth arriving customer;
- Wn, the waiting time of the nth arriving customer to the GI/G/1 system.
Since an upward and a downward shift in rho are associated with a deterioration and an improvement (respectively) of the quality of service, the timely detection of these changes is an imperative requirement, hence, begging for the use of ARL-unbiased charts (Pignatiello et al., 1995), in the sense that they detect any shifts in the traffic intensity sooner than they trigger a false alarm.
In this paper, we focus on the design of these type of charts for the traffic intensity of the three single server queues mentioned above.