On ARL-Unbiased Charts to Monitor the Traffic Intensity of a Single Server Queue

Abstract

We know too well that the effective operation of a queueing system requires maintaining the traffic intensity ρ at a target value ρ ₀.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 (Oper Res 20:955–966, 1972) or more recently in Chen and Zhou (Technometrics 57:245–256, 2015).

For all intents and purposes, this chapter focus on three control statistics chosen by Morais and Pacheco (Seq Anal 35:536–559, 2016) for their simplicity, recursive and Markovian character. Since an upward and a downward shift in ρ 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., The performance of control charts for monitoring process dispersion. In: 4th industrial engineering research conference, pp 320–328, 1995), in the sense that they detect any shifts in the traffic intensity sooner than they trigger a false alarm.

In this chapter, we focus on the design of these type of charts for the traffic intensity of the three single server queues mentioned above.

Appendix

If the service times of an M∕G∕1 queueing system have an Erlang distribution with $k\ (k \in \mathbb {N})$ phases and probability density function (p.d.f.) given by

$$\displaystyle \begin{aligned} f_S(s) = (k \mu)^k \, s^{k-1} \, e^{-k \mu s} / (k-1)!, \quad s \geq 0, \end{aligned}$$

then

$$\displaystyle \begin{aligned} \alpha_i = {k+i-1 \choose k-1} \left( \frac{\rho}{k+\rho} \right)^i \left( \frac{k}{k+\rho}\right)^k, \quad i \in \mathbb{N}_0 \end{aligned} $$

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(Feller 1971, p. 57). In other words, Y  has a negative binomial distribution with parameters k and k(k + ρ)⁻¹, when we are dealing with the M∕E _k ∕1 queueing system.

If the GI∕M∕1 queueing system is associated with interarrival times with an Erlang distribution with density

$$\displaystyle \begin{aligned} f_A(a) = (k \lambda)^k \, a^{k-1} \, e^{-k \lambda a} / (k-1)!, \quad a \geq 0, \end{aligned}$$

Morais and Pacheco (2016) adds that

$$\displaystyle \begin{aligned} \hat{\alpha}_i = {k+i-1 \choose k-1} \left( \frac{k^{-1}}{k^{-1}+\rho} \right)^i \left( \frac{\rho}{k^{-1}+\rho}\right)^k, \quad i \in \mathbb{N}_0. \end{aligned} $$

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This is to say that Y  has a negative binomial distribution with parameters k and ρ (k ⁻¹ + ρ)⁻¹, for the E _k ∕M∕1 queue.

When it comes to the GI∕G∕1 queueing system, the results derived by Nadarajah and Kotz (2005), for the c.d.f. and p.d.f. of a linear combination (αX + βY ) of exponential (X) and gamma (Y ) independent r.v. (with α > 0), come in handy.

For the M∕M∕1 queueing system with arrival rate λ = 1∕E(A) and service rate μ = 1∕E(S), Morais and Pacheco (2016) wrote

$$\displaystyle \begin{aligned} F_{S-A}(x) = \left\{ \begin{array}{l} \frac{\mu \, e^{\lambda x}}{\lambda + \mu}, \quad x \leq 0 \\ 1- \frac{\lambda \, e^{-\mu x}}{\lambda + \mu}, \quad x > 0. \end{array} \right. \end{aligned} $$

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Similar calculations led Morais and Pacheco (2016) to conclude that:

$$\displaystyle \begin{aligned} F_{S-A}(x) = \left\{ \begin{array}{ll} e^{\lambda x} \, \left( \frac{k\mu}{k\mu + \lambda} \right)^k, \quad x \leq 0\\ F_{Gamma(k,k\mu)}(x) + e^{\lambda x} \, \left( \frac{k\mu}{k\mu + \lambda} \right)^k \, \bar{F}_{Gamma(k,k\mu+\lambda)}(x), \quad x > 0, \end{array} \right. \end{aligned} $$

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for the M∕E _k ∕1 queueing system; and

$$\displaystyle \begin{aligned} F_{S-A}(x) = \left\{ \begin{array}{ll} \bar{F}_{Gamma(k,k\lambda)}(-x) \\ \quad \quad - e^{-\mu x} \, \left( \frac{k\lambda}{k\lambda + \mu} \right)^k \, \bar{F}_{Gamma(k,k\lambda+\mu)}(-x), \quad x \leq 0 \\ 1 - e^{-\mu x} \, \left( \frac{k\lambda}{k\lambda + \mu} \right)^k, \quad x > 0, \end{array} \right. \end{aligned} $$

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for the E _k ∕M∕1 system.