## 1 Introduction

Queueing theory (QT) is fundamentally devoted to the modelling of systems where customers (eventually) wait for service and congestion occurs due to the stochastic character of the arrival process and of the service times [12, p. 483].
In this paper, we assume that customers come from an infinite source, arrive individually to an infinite capacity queueing system with *m* parallel servers providing service according to the first come first served (FCFS) discipline.

Regarding the *arrival process*, we suppose that the interarrival times are independent and identically distributed (i.i.d.) positive random variables (r.v.) to *U* with common cumulative distribution function (c.d.f.)

The time elapsed while a customer is being served is called *service time*.
We assume that the service times are i.i.d. positive r.v. to *V* with common c.d.f.

One of the simplest queueing system to analyze is the *M*, and *GI* (or *G*) stand for exponential (Memoryless, Markovian) and
general independent interarrival (or service) times, respectively.
Accordingly, if the single-server queue admits:

- •Markovian arrivals, but the service process is governed by a general distribution: then we are dealing with an
queueing system.$M/G/1$ - •General interarrival times and exponentially distributed service times: then we are considering a
system.$GI/M/1$ - •Interarrival and service times with general distributions: then we are discussing a
queueing system.$GI/G/1$

When we assess a queueing system, it is critical to consider the total workload submitted to the *m* servers in the interval *i*-th arrival, then

can be thought as an average measure of congestion in the system
at each time *t*.
Thus,

is called the (long-run) traffic intensity of the queueing system and represents the load offered to each server if the work is divided equally among servers [13, p. 6]. If a queueing system has an unlimited waiting room, then

Bearing in mind that detecting persistent shifts in the traffic intensity is vitally important while dealing with queueing systems, it is remarkable that we have to leap to the early 1970s to meet the authors of a seminal work proposing a control chart for the traffic intensity of the

The approximate control technique proposed by Bhat and Rao [4] was inspired by Shewhart quality control charts and provides a method of regulating the traffic intensity of those queueing systems by observing only the number of customers in the system at embedded Markov points:

- •
, the number of customers in the${X}_{n}$ queueing system immediately after the$M/G/1$ *n*-th departure, - •
, the number of customers in the${\widehat{X}}_{n}$ system, as seen by the$GI/M/1$ *n*-th arriving customer.

The traffic intensity is deemed out-of-control if the control statistic exceeds (resp. does not exceed) the upper (resp. lower) control limit

It is imperative to mention that

- •
, the waiting time of the${W}_{n}$ *n*-th arriving customer to a system,$GI/G/1$

appears to have been used for the first time – albeit as a building block of a cumulative sum (CUSUM) control statistic – by Kim, Alexopoulos, Tsui and Wilson [8], who formulated a distribution-free tabular cumulative sum (DFTC) chart to monitor a discrete time stochastic process, thus generalizing the conventional CUSUM chart for i.i.d. normal r.v.

It is also important to allude to the two cumulative sum (CUSUM) schemes applied by Chen and Zhou
[7] in order to efficiently monitor the performance of *partial sampling scheme*, which only observes the number of customers left in the system, and the *complete sampling scheme*, which records each event type (arrival or departure) and the corresponding time epoch, leading to the CUSUM-P and the CUSUM-C charts, respectively.

For an unintentionally biased, most likely incomplete and somewhat overlapping reviews of the monitoring of the traffic intensity of queueing systems and other stochastic systems the reader is referred for instance to [10], [14, Section 1.7] and [15].

The remainder of this paper is organized as follows.
In Section 2, we briefly refer to three *short-memory charts* for ρ, namely the

## 2 Short-Memory Charts for ρ

The control statistics

Control statistics for different queueing systems.

System | Control statistic | Chart |

In
Table 1,
the *increments*

- •
is the number of customers arriving during the service of the${Y}_{n+1}$ -st customer,$(n+1)$ - •
is the number of potential customers served during the${\widehat{Y}}_{n+1}$ -st interarrival time period,$(n+1)$ - •
:$({V}_{n+1},{U}_{n+1})$ is the service time of the${V}_{n+1}$ *n*-th customer, and denotes the time between the arrivals of customers${U}_{n+1}$ *n*and ,$(n+1)$ ,$n\in {\mathbb{N}}_{0}$

with

The p.f. of *Y* and

- •the exponential distribution
,$(M)$ - •the Erlang distribution
with shape parameter$({E}_{k})$ *k*, - •the hypoexponential distribution
,$({\text{Hypo}}_{k})$ - •the hyperexponential distribution
.$({H}_{k})$

The *short-memory charts* because

The performance of these control charts is usually assessed by calculating the average run length (ARL), that is, the average number of samples taken before a signal is triggered. The ARL should be large (resp. short) when the traffic intensity is on-target (resp. off-target).

Santos [14, Chapter 2]
and Santos, Morais and Pacheco
[15]
capitalize on the Markov chain approach [5]
to compute several RL related measures for the

- •out-of-control scenarios,
- •interarrival and service time distributions,

by considering not only the

To provide a more thorough answer to the question

When it comes to detection speed, does it pay off to use the control statisticthat requires more bookkeeping than the discrete statistics ${W}_{n}$ and ${X}_{n}$ ? ${\widehat{X}}_{n}$

Santos [14, Chapter 2] and Santos, Morais and Pacheco [15] considered three out-of-control scenarios because Chen and Zhou [7] claim that, in the context of detecting queueing congestion, decreases of the service rate and/or increases of the arrival rate are of most interest.

The first two out-of-control scenarios refer to increases in the traffic intensity from its target value

- (i)decreases in the service rate while the arrival rate is constant:
${\mu}_{1}={\mu}_{0}\frac{{\rho}_{0}}{{\rho}_{1}}<{\mu}_{0}\mathit{\hspace{1em}}\text{and}\mathit{\hspace{1em}}{\lambda}_{1}={\lambda}_{0},$ - (ii)increases in the arrival rate while the service rate remains unchanged:
${\lambda}_{1}={\lambda}_{0}\frac{{\rho}_{1}}{{\rho}_{0}}>{\lambda}_{0}\mathit{\hspace{1em}}\text{and}\mathit{\hspace{1em}}{\mu}_{1}={\mu}_{0}.$

In the third out-of-control scenario,

- (iii)the arrival rate increases and the service rate decreases proportionally:
${\lambda}_{1}=\sqrt{\frac{{\rho}_{1}}{{\rho}_{0}}}{\lambda}_{0}>{\lambda}_{0}\mathit{\hspace{1em}}\text{and}\mathit{\hspace{1em}}{\mu}_{1}=\sqrt{\frac{{\rho}_{0}}{{\rho}_{1}}}{\mu}_{0}<{\mu}_{0}.$

Santos, Morais and Pacheco [15] provided evidence that:

- •the
-chart outperforms the${W}_{n}$ -chart (resp.${X}_{n}$ -chart) when it comes to the detection of increases in the traffic intensity of${\widehat{X}}_{n}$ ,$M/M/1$ ,$M/{E}_{2}/1$ and$M/{\text{Hypo}}_{2}/1$ (resp.$M/{H}_{2}/1$ ,$M/M/1$ ,${E}_{2}/M/1$ and${\text{Hypo}}_{2}/M/1$ ) queues under scenario (i).${H}_{2}/M/1$ - •the
-chart (resp.${X}_{n}$ -chart) proved to be faster than the${\widehat{X}}_{n}$ -chart in the detection of increases in the traffic intensity under scenario (ii), for all the queueing systems mentioned above.${W}_{n}$ - •when the increase in ρ is attributable to an increase in λ and a proportional decrease in μ, i.e., we are dealing with scenario (iii), adopting the waiting time of an arriving customer as a control statistic instead of
does not necessarily payoff in terms of the ARL performance.${X}_{n}$

These are quite surprising results because the extra *bookkeeping* associated with the collection of the waiting times of the arriving customers would suggest swifter detections by the

The mixed character of

## 3 Long-Memory Charts for ρ

Despite their popularity and simplicity, charts that only use the last observed value of their control statistics to trigger a signal have a well-known limitation: they are not very effective in the detection of small and moderate shifts.

To overcome this problem, the control statistics should explicitly rely on the information contained in the most recent and the past collected samples of the process.
That is the case of the CUSUM-P and CUSUM-C charts proposed by Chen and Zhou
[7].
The complexity of the control statistics of the CUSUM-P and CUSUM-C charts for the traffic intensity, when compared to the ones in the previous section, led us to term them *long-memory charts*.
It also prompted us to restrict ourselves to the

What follows is a brief description of the CUSUM charts proposed by Chen and Zhou
[7]
to monitor the traffic intensity of an

The state of the *complete sampling scheme* and the CUSUM-C chart.

These authors also call our attention to the fact that due to practical constraints, sometimes it is not possible to record all this information.
For instance, in a production line without individual tracking capability, only the number of units in the system can be obtained using proximity sensors, and the waiting time of each unit is not available.
To account for concrete situations such as this, Chen and Zhou
[7]
suggested a *partial sampling scheme* associated only with records of the number of customers left in the

In the partial sampling scheme, the observations correspond to realizations of the number of customers left in the system at each departure epoch in the

where *k* if the control statistic

The observed value of the control statistic of the CUSUM-C chart to monitor the traffic intensity of an

where *N*-th departing customer; and *N*-th departure.
The CUSUM-C chart triggers a signal at sample *k* if the control statistic

The approximate ARL of the CUSUM-P and CUSUM charts, used to monitor a system in steady-state operation, can be found in [7]. Even though we assume that we start with an empty queueing system and monitor its traffic intensity in the transient state, these approximations are useful in order to check our results.

We can use Monte Carlo simulation to determine the upper control limits of these two types of CUSUM charts such that its in-control ARL meets a pre-specified value and estimate out-of-control RL related measures such as the expected value, standard deviation, coefficient of variation and

## 4 Comparing Short and Long-Memory Charts

By taking into account more information, the CUSUM-C chart is able to detect shifts in the service rate and/or in the arrival rate separately.
Through several numerical examples, Chen and Zhou
[7]
showed, for instance, that the ARL performance of the CUSUM-C chart tends to be better than the one of:
the CUSUM-P and the *WZ* charts;
what Chen, Yuan and Zhou
[6]
called the *nL*-chart, based on non-overlapping sums of *n* consecutive observations of the number of customers at departure epochs;
or the generalized likelihood ratio chart under partial sampling.

The main goal of this section is to compare the RL performance of the

For simplicity’s sake, we consider

- •
under scenario (i),$({\lambda}_{1},{\mu}_{1})=({\lambda}_{0},{\mu}_{0}{\rho}_{0}/{\rho}_{1})$ - •
under scenario (ii),$({\lambda}_{1},{\mu}_{1})=({\lambda}_{0}{\rho}_{1}/{\rho}_{0},{\mu}_{0})$ - •
under scenario (iii).$({\lambda}_{1},{\mu}_{1})=(\sqrt{{\rho}_{1}/{\rho}_{0}}{\lambda}_{0},\sqrt{{\rho}_{0}/{\rho}_{1}}{\mu}_{0})$

The performance of these short and long-memory charts is assessed for two different target values of the traffic intensity:

When it comes to Monte Carlo simulations, we consider that systems start from an empty queue and use
*Matlab* [16]
to simulate *Matlab*’s procedure rng.
Each run is associated with fixed arrival and service rates and a maximum of

While comparing the CUSUM charts associated with complete and partial sampling schemes, we made sure that the design parameters verify

Due to the inherent limitations of the Monte Carlo simulations and the fact that we considered upper control limits with only two decimal places, the CUSUM charts are roughly matched in-control to the

The estimates of the RL percentage points, ARL, SDRL (standard deviation of the RL), CVRL (coefficient of variation of the RL) of the

RL percentage points | |||||||||||||

ρ | ARL | SDRL | CVRL | Scenario / UCL | |||||||||

0.3 | 6 | 21 | 41 | 108 | 257 | 512 | 849 | 1104 | 1696 | 370.000 | 367.945 | 0.9944 | IC / 5.72998 |

0.33 | 4 | 13 | 25 | 64 | 150 | 298 | 494 | 643 | 987 | 215.823 | 213.828 | 0.9908 | (i) |

0.36 | 3 | 9 | 16 | 41 | 96 | 191 | 316 | 410 | 630 | 138.272 | 136.341 | 0.9860 | |

0.45 | 2 | 4 | 7 | 16 | 37 | 73 | 119 | 155 | 237 | 52.795 | 51.049 | 0.9669 | |

0.33 | 5 | 18 | 34 | 90 | 214 | 426 | 706 | 917 | 1409 | 307.732 | 305.512 | 0.9928 | (ii) |

0.36 | 5 | 16 | 29 | 76 | 180 | 358 | 593 | 770 | 1183 | 258.783 | 256.398 | 0.9908 | |

0.45 | 4 | 11 | 20 | 49 | 114 | 224 | 371 | 481 | 738 | 162.600 | 159.726 | 0.9823 | |

0.33 | 5 | 15 | 29 | 75 | 179 | 355 | 588 | 765 | 1175 | 256.692 | 254.588 | 0.9918 | (iii) |

0.36 | 4 | 12 | 22 | 55 | 130 | 258 | 427 | 555 | 852 | 186.694 | 184.550 | 0.9885 | |

0.45 | 3 | 7 | 11 | 27 | 62 | 122 | 200 | 260 | 399 | 88.285 | 86.061 | 0.9748 | |

0.5 | 8 | 24 | 43 | 110 | 258 | 511 | 846 | 1099 | 1687 | 370.000 | 365.260 | 0.9872 | IC / 7.74703 |

0.55 | 6 | 14 | 25 | 60 | 139 | 273 | 450 | 584 | 896 | 198.075 | 193.533 | 0.9771 | (i) |

0.6 | 4 | 10 | 17 | 38 | 85 | 165 | 271 | 351 | 537 | 120.062 | 115.735 | 0.9640 | |

0.75 | 3 | 5 | 8 | 15 | 31 | 58 | 94 | 122 | 185 | 43.073 | 39.394 | 0.9146 | |

0.55 | 7 | 19 | 34 | 83 | 193 | 381 | 629 | 817 | 1253 | 276.084 | 270.876 | 0.9811 | (ii) |

0.6 | 7 | 16 | 27 | 65 | 148 | 290 | 477 | 619 | 949 | 210.484 | 204.819 | 0.9731 | |

0.75 | 6 | 12 | 17 | 35 | 75 | 143 | 234 | 302 | 460 | 105.328 | 98.430 | 0.9345 | |

0.55 | 6 | 17 | 29 | 71 | 163 | 321 | 530 | 688 | 1056 | 233.068 | 228.204 | 0.9791 | (iii) |

0.6 | 5 | 13 | 21 | 49 | 111 | 216 | 356 | 462 | 707 | 157.377 | 152.429 | 0.9686 | |

0.75 | 4 | 8 | 11 | 23 | 47 | 89 | 145 | 187 | 284 | 65.631 | 60.624 | 0.9237 |

CUSUM-P chart for an

RL percentage points | |||||||||||||

ρ | ARL | SDRL | CVRL | Scenario / UCL | |||||||||

0.3 | 29 | 54 | 76 | 140 | 280 | 508 | 796 | 1020 | 1572 | 375.382 | 330.094 | 0.8794 | IC / 0.72 |

0.33 | 24 | 42 | 56 | 93 | 168 | 291 | 452 | 574 | 861 | 219.666 | 179.222 | 0.8159 | (i) |

0.36 | 20 | 32 | 42 | 69 | 117 | 187 | 286 | 363 | 551 | 145.907 | 110.464 | 0.7571 | |

0.45 | 12 | 20 | 26 | 39 | 58 | 86 | 119 | 144 | 202 | 67.184 | 40.462 | 0.6023 | |

0.33 | 25 | 42 | 57 | 95 | 168 | 292 | 456 | 577 | 840 | 219.875 | 176.063 | 0.8007 | (ii) |

0.36 | 20 | 34 | 43 | 68 | 114 | 186 | 279 | 356 | 507 | 142.217 | 103.934 | 0.7308 | |

0.45 | 13 | 22 | 27 | 39 | 59 | 87 | 121 | 146 | 205 | 68.226 | 40.233 | 0.5897 | |

0.33 | 26 | 42 | 56 | 93 | 166 | 289 | 452 | 578 | 830 | 218.528 | 175.416 | 0.8027 | (iii) |

0.36 | 20 | 36 | 44 | 70 | 116 | 195 | 288 | 361 | 539 | 147.223 | 109.632 | 0.7447 | |

0.45 | 14 | 21 | 27 | 39 | 58 | 88 | 122 | 147 | 207 | 68.352 | 40.92 | 0.5987 | |

0.5 | 27 | 52 | 76 | 140 | 275 | 501 | 801 | 1014 | 1518 | 372.462 | 325.127 | 0.8729 | IC / 0.83 |

0.55 | 23 | 38 | 50 | 85 | 149 | 261 | 403 | 524 | 778 | 196.523 | 160.064 | 0.8145 | (i) |

0.6 | 18 | 29 | 38 | 60 | 98 | 160 | 238 | 302 | 436 | 122.823 | 89.350 | 0.7275 | |

0.75 | 12 | 19 | 23 | 33 | 49 | 72 | 100 | 121 | 170 | 56.927 | 33.108 | 0.5816 | |

0.55 | 18 | 33 | 45 | 78 | 143 | 263 | 418 | 540 | 802 | 196.361 | 168.897 | 0.8601 | (ii) |

0.6 | 14 | 25 | 33 | 54 | 94 | 159 | 241 | 309 | 452 | 120.774 | 94.629 | 0.7835 | |

0.75 | 10 | 16 | 20 | 30 | 45 | 69 | 98 | 120 | 175 | 53.772 | 33.996 | 0.6322 | |

0.55 | 13 | 24 | 35 | 69 | 142 | 271 | 445 | 584 | 843 | 200.622 | 185.541 | 0.9248 | (iii) |

0.6 | 10 | 19 | 27 | 47 | 91 | 167 | 262 | 335 | 532 | 124.485 | 110.782 | 0.8899 | |

0.75 | 7 | 12 | 15 | 25 | 42 | 67 | 101 | 127 | 188 | 51.779 | 38.271 | 0.7391 |

CUSUM-C chart: in-control and out-of-control

RL percentage points | |||||||||||||

ρ | ARL | SDRL | CVRL | Scenario / UCL | |||||||||

22 | 46 | 69 | 130 | 264 | 496 | 797 | 1030 | 1578 | 368.209 | 335.970 | 0.9124 | IC / 1.26 | |

0.33 | 17 | 29 | 39 | 63 | 108 | 183 | 276 | 358 | 526 | 138.873 | 108.083 | 0.7783 | (i) |

0.36 | 13 | 21 | 26 | 40 | 63 | 98 | 144 | 180 | 263 | 76.650 | 52.888 | 0.6900 | |

0.45 | 7 | 11 | 14 | 20 | 29 | 40 | 54 | 64 | 86 | 32.132 | 16.724 | 0.5205 | |

0.3 | 39 | 62 | 81 | 141 | 272 | 496 | 790 | 1016 | 1552 | 370.602 | 324.275 | 0.8750 | IC / 1.27 |

0.33 | 33 | 44 | 54 | 77 | 122 | 195 | 284 | 353 | 518 | 150.190 | 101.898 | 0.6785 | (ii) |

0.36 | 28 | 36 | 42 | 55 | 76 | 109 | 152 | 186 | 257 | 88.974 | 48.447 | 0.5445 | |

0.45 | 21 | 26 | 29 | 34 | 41 | 51 | 63 | 71 | 89 | 44.031 | 14.111 | 0.3205 | |

0.3 | 33 | 55 | 78 | 139 | 272 | 495 | 792 | 984 | 1509 | 368.550 | 320.474 | 0.8696 | IC / 0.96 |

0.33 | 28 | 43 | 54 | 83 | 140 | 229 | 344 | 434 | 628 | 175.785 | 130.548 | 0.7427 | (iii) |

0.36 | 23 | 35 | 43 | 63 | 93 | 148 | 216 | 267 | 379 | 115.358 | 76.753 | 0.6653 | |

0.45 | 15 | 22 | 27 | 36 | 49 | 67 | 90 | 106 | 142 | 54.743 | 26.843 | 0.4903 | |

26 | 49 | 72 | 134 | 267 | 497 | 786 | 999 | 1601 | 367.008 | 330.097 | 0.8994 | IC / 1.27 | |

0.55 | 17 | 28 | 38 | 64 | 109 | 181 | 277 | 362 | 535 | 139.107 | 109.501 | 0.7872 | (i) |

0.6 | 14 | 21 | 27 | 41 | 64 | 100 | 142 | 177 | 255 | 77.263 | 51.806 | 0.6705 | |

0.75 | 7 | 12 | 14 | 20 | 29 | 41 | 55 | 65 | 88 | 32.544 | 17.090 | 0.5252 | |

0.5 | 40 | 63 | 83 | 141 | 275 | 507 | 795 | 999 | 1563 | 373.974 | 325.184 | 0.8695 | IC / 1.28 |

0.55 | 30 | 43 | 53 | 77 | 123 | 199 | 296 | 366 | 531 | 153.551 | 108.713 | 0.7080 | (ii) |

0.6 | 24 | 34 | 41 | 54 | 76 | 109 | 153 | 188 | 268 | 88.858 | 50.717 | 0.5708 | |

0.75 | 15 | 22 | 26 | 32 | 41 | 52 | 64 | 72 | 87 | 42.995 | 15.207 | 0.3537 | |

0.5 | 32 | 56 | 80 | 139 | 271 | 486 | 786 | 1004 | 1546 | 366.764 | 324.445 | 0.8846 | IC / 0.96 |

0.55 | 25 | 41 | 53 | 85 | 141 | 238 | 363 | 450 | 679 | 181.432 | 138.209 | 0.7618 | (iii) |

0.6 | 21 | 33 | 42 | 60 | 94 | 143 | 210 | 257 | 375 | 112.517 | 73.572 | 0.6539 | |

0.75 | 13 | 21 | 25 | 35 | 48 | 67 | 89 | 105 | 144 | 53.679 | 27.326 | 0.5091 |

Before discussing the results for each target value

For

comprises explicit information on the service time

and increase with ρ when the arrival (or service) rate is constant
[11, pp. 80–81].
As a consequence, if ρ increases due to a downward shift in the service rate while the arrival rate is fixed, then the values of

Under scenario (ii), the

For

Judging by the results in
Table 3,
we can add that the performance of the CUSUM-P chart is approximately the same under all the three out-of-control scenarios for a fixed

Interestingly, the difference between the RL performances of CUSUM-P and CUSUM-C charts seems to be more accentuated under scenarios (i) and (ii). Moreover, under scenario (iii), the ARL profiles in Figure 1 and the results in Tables 3 and 4 indicate that replacing the CUSUM-P chart with the CUSUM-C chart does not lead to a significant improvement in the ARL performance.

## 5 Concluding Remarks

The main goal of this paper was to compare short and long-memory control charts used to detect increases in the traffic intensity of single server queues, under different out-of-control scenarios.
We focused on two different approaches to calculate RL performance metrics:
the Markov chain approach, to compute the RL performance of the

Even though the CUSUM-C chart showed the best overall ARL performance, the associated *bookkeeping* is a major disadvantage since it means that we need to keep track not only of the number of customers in the system after each departure, but also of all the arrival and departure epochs.
An additional drawback of this chart follows from the fact that the choice of its design parameters requires *a priori* knowledge of the off-target values of λ and μ we want to detect, thus making it less appealing.

The quality practitioner should also be aware of the fact that the ARL performance of the CUSUM-C chart is very sensitive to the different out-of-control scenarios and to the relative position of the design parameters

Curiously, a comparison between the performances of the CUSUM-P and

Since downward (resp. upward) shifts in the traffic intensity can correspond to a decreasing (resp. increasing) interest in the offered services
[9],
it is important to promptly detect both increases and decreases in ρ.
This calls for an ARL-unbiased chart such as the one based on *ARL-unbiased* versions of the CUSUM-C and CUSUM-P charts, such that:
their in-control ARL take a pre-stipulated value

Another obvious direction for future work refers to the derivation of the CUSUM-P and CUSUM-C control charts to monitor the traffic intensity of queueing systems other than the

We are greatly indebted to the Referee who selflessly devoted invaluable time to scrutinize this work.

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