Comparing Short and Long-Memory Charts to Monitor the Traffic Intensity of Single Server Queues

Marta Santos 1 , Manuel Cabral Morais 2 ,  and António Pacheco 3
  • 1 Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, 1049-001, Lisboa, Portugal
  • 2 Instituto Superior Técnico, Universidade de Lisboa, Department of Mathematics & CEMAT (Center for Computational and Stochastic Mathematics), Lisboa, Portugal
  • 3 Instituto Superior Técnico, Universidade de Lisboa, Department of Mathematics & CEMAT (Center for Computational and Stochastic Mathematics), Lisboa, Portugal
Marta Santos, Manuel Cabral Morais
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  • Department of Mathematics & CEMAT (Center for Computational and Stochastic Mathematics), Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, 1049-001, Lisboa, Portugal
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and António Pacheco
  • Department of Mathematics & CEMAT (Center for Computational and Stochastic Mathematics), Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, 1049-001, Lisboa, Portugal
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Abstract

The traffic intensity (ρ) is a vital parameter of queueing systems because it is a measure of the average occupancy of a server. Consequently, it influences their operational performance, namely queue lengths and waiting times. Moreover, since many computer, production and transportation systems are frequently modelled as queueing systems, it is crucial to use control charts to detect changes in ρ. In this paper, we pay particular attention to control charts meant to detect increases in the traffic intensity, namely: a short-memory chart based on the waiting time of the n-th arriving customer; two long-memory charts with more sophisticated control statistics, and the two cumulative sum (CUSUM) charts proposed by Chen and Zhou (2015). We confront the performances of these charts in terms of some run length related performance metrics and under different out-of-control scenarios. Extensive results are provided to give the quality control practitioner a concrete idea about the performance of these charts.

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.) FU(u) and such that E(U)=λ-1, where λ denotes the long-run (customer) arrival rate.

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. FV(v) and expected value E(V)=μ-1, so that μ denotes the (customer) service rate. We also suppose that the service times are independent of the interarrival times.

One of the simplest queueing system to analyze is the M/M/1 queue, where interarrival times and service times are exponentially distributed and there is a single-server in the system. Here, the acronym M/M/1 follows the usual notation for queueing systems, where the code letters 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:

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

When we assess a queueing system, it is critical to consider the total workload submitted to the m servers in the interval [0,t] due to the N(t) arrivals in that period [13, p. 5]. If Si represents the service time of the i-th arrival, then

ρ(t)=E[1tmi=1N(t)Si]

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

ρ=limtρ(t)

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

ρ=λmμ.

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 M/G/1 and GI/M/1 queueing systems. Indeed, Bhat [1] proposed a parameter regulation technique for time homogeneous stochastic processes that was specialized in [4] to control the traffic intensity of M/G/1 and GI/M/1 queues [2]; Bhat and Rao [4] added that the results are obtained through a direct application of the work reported in [3].

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:

  1. Xn, the number of customers in the M/G/1 queueing system immediately after the n-th departure,
  2. X^n, the number of customers in the GI/M/1 system, as seen by the 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 cu (resp. cl) longer than a pre-assigned number du (resp. dl) of consecutive transitions. Thus, unlike in traditional control charts, the interval [cl,cu] can be considered as forming a warning zone (WZ) as values beyond these control limits are not immediately responsible for signals.

It is imperative to mention that

  1. Wn, the waiting time of the n-th arriving customer to a GI/G/1 system,

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 M/M/1 queues. The 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 Wn-chart, thoroughly discussed in [14] and [15]. Section 3 is devoted to the description of the CUSUM-P and CUSUM-C charts. In Section 4, we compare the performance of the Wn and the CUSUM-P and CUSUM-C charts under several out-of-control scenarios. Clear conclusions and recommendations are given in Section 5.

2 Short-Memory Charts for ρ

The control statistics Xn, X^n and Wn have been used by several authors to monitor the traffic intensity because of their simplicity, recursive and Markovian character plainly suggested by their expressions in Table 1. These properties prove to be decisive in characterizing the performance of the associated control charts.

Table 1

Control statistics for different queueing systems.

SystemControl statisticChart
M/G/1Xn+1=max{0,Xn-1}+Yn+1Xn-chart
GI/M/1X^n+1=max{0,X^n+1-Y^n+1}X^n-chart
GI/G/1Wn+1=max{0,Wn+Vn+1-Un+1}Wn-chart

In Table 1, the incrementsYn+1, -Y^n+1, and Vn+1-Un+1 are characterized as follows:

  1. Yn+1 is the number of customers arriving during the service of the (n+1)-st customer,
  2. Y^n+1 is the number of potential customers served during the (n+1)-st interarrival time period,
  3. (Vn+1,Un+1): Vn+1 is the service time of the n-th customer, and Un+1 denotes the time between the arrivals of customers n and (n+1), n0,

with

Yni.i.d.Y,n,Y^ni.i.d.Y^,n,Vn+1-Un+1i.i.d.V-U,n0.

The p.f. of Y and Y^ and the c.d.f. of V-U can be found in [15, pp. 19–20] for a handful of different distributions for the interarrival and service times such as:

  1. the exponential distribution (M),
  2. the Erlang distribution (Ek) with shape parameter k,
  3. the hypoexponential distribution (Hypok),
  4. the hyperexponential distribution (Hk).

The Xn, X^n and Wn charts trigger a signal if their control statistics exceed a critical level and play a vital role in the detection of increases in the traffic intensity of M/G/1, GI/M/1 and GI/G/1 queueing systems, respectively. Even though their control statistics have a recursive character, we termed these charts short-memory charts because Xn, X^n and Wn refer solely to a customer-arrival/departure epoch as opposed to several such epochs.

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 Xn, X^n and Wn charts, and extend the analysis in [10] to a broader set of

  1. out-of-control scenarios,
  2. interarrival and service time distributions,

by considering not only the M/M/1, M/E2/1 and E2/M/1 queues but also the M/H2/1, H2/M/1, M/Hypo2/1 and Hypo2/M/1 queueing systems.

To provide a more thorough answer to the question

When it comes to detection speed, does it pay off to use the control statistic Wn that requires more bookkeeping than the discrete statistics Xn and 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 ρ0=λ0μ0 to ρ1=λ1μ1 due to:

  1. (i)decreases in the service rate while the arrival rate is constant:
    μ1=μ0ρ0ρ1<μ0andλ1=λ0,
  2. (ii)increases in the arrival rate while the service rate remains unchanged:
    λ1=λ0ρ1ρ0>λ0andμ1=μ0.

In the third out-of-control scenario,

  1. (iii)the arrival rate increases and the service rate decreases proportionally:
    λ1=ρ1ρ0λ0>λ0andμ1=ρ0ρ1μ0<μ0.

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

  1. the Wn-chart outperforms the Xn-chart (resp. X^n-chart) when it comes to the detection of increases in the traffic intensity of M/M/1, M/E2/1, M/Hypo2/1 and M/H2/1 (resp. M/M/1, E2/M/1, Hypo2/M/1 and H2/M/1) queues under scenario (i).
  2. the Xn-chart (resp. X^n-chart) proved to be faster than the Wn-chart in the detection of increases in the traffic intensity under scenario (ii), for all the queueing systems mentioned above.
  3. 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 Xn does not necessarily payoff in terms of the ARL performance.

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 Wn-chart. However, keep in mind the following important characteristics of the Wn-chart: it can be used to monitor the traffic intensity of any GI/G/1 system whereas the Xn-chart (resp. X^n-chart) is only meant to be used to control the traffic intensity of M/G/1 (resp. GI/M/1) queues.

The mixed character of Wn allows us to set the associated chart for any pre-specified (and reasonably large) in-control (IC) ARL, unlike the short-memory charts with discrete control statistics Xn and X^n; the RL performance of the Wn-chart depends upon both λ and μ, in contrast with the RL performance of the Xn and X^n charts that depend exclusively on ρ=λμ.

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 M/M/1 queueing system.

What follows is a brief description of the CUSUM charts proposed by Chen and Zhou [7] to monitor the traffic intensity of an M/M/1 system. Please note that Chen and Zhou [7] made it clear that these CUSUM charts can be extended to more general queues and that they considered this simple queueing system to avoid potential distraction and confusion caused by additional required knowledge in QT needed for other queueing systems.

The state of the M/M/1 system only undergoes a change when there is an arrival or a departure [7]. Consequently, it is sufficient to record the event type (that is, an arrival or a departure) and the corresponding event time (i.e., the arrival or departure time) to completely determine the sample path of the process, thus leading to what Chen and Zhou [7] called 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 M/M/1 system at departure epochs and leading to what they called a CUSUM-P chart.

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 M/M/1 system, say x1,x2,,xk,. The control statistic of the CUSUM-P chart at the (k+1)-st departure epoch depends not only on xk and xk+1 but also on the target traffic intensity ρ0 and the out-of-control traffic intensity ρ1 that we are most interested in detecting. It has a recursive character and its observed value is

gk+1=max{0,gk+ln[ρ1(1+ρ0)ρ0(1+ρ1)](xk+1-xk+1-𝕀xk=0)-ln(1+ρ11+ρ0)},k,

where g0=0 and ρ1>ρ0. The CUSUM-P chart is responsible for a signal at sample k if the control statistic gk exceeds its upper control limit, suggesting that the traffic intensity has suffered an upward shift from its target value ρ0.

The observed value of the control statistic of the CUSUM-C chart to monitor the traffic intensity of an M/M/1 system is given by

hk+1=max{0,hk+ln(μ1μ0)+(xk+1-xk+1)ln(λ1λ0)-(μ1-μ0)sk+1-(λ1-λ0)(tk+1-tk)},k,

where h0=0; the design parameters are (λ1,μ1) and verify ρ1=λ1μ1>ρ0=λ0μ0; sN is the service time of the N-th departing customer; and tN is the overall system time elapsed until the N-th departure. The CUSUM-C chart triggers a signal at sample k if the control statistic hk exceeds its UCL. This signal suggests that the arrival or the service rate have changed from their target values λ0 and μ0.

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 p×100% percentage point of the RL.

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 Wn-chart with the one of the CUSUM-P and CUSUM-C charts under the out-of-control scenarios (i), (ii) and (iii) described in Section 2.

For simplicity’s sake, we consider λ0=ρ0 and μ0=1. Moreover, we fix the design parameter ρ1=1.1ρ0 for the CUSUM-P charts and the corresponding set of design parameters for the CUSUM-C charts are:

  1. (λ1,μ1)=(λ0,μ0ρ0/ρ1) under scenario (i),
  2. (λ1,μ1)=(λ0ρ1/ρ0,μ0) under scenario (ii),
  3. (λ1,μ1)=(ρ1/ρ0λ0,ρ0/ρ1μ0) under scenario (iii).

The performance of these short and long-memory charts is assessed for two different target values of the traffic intensity: ρ0=0.3,0.5. The latter target value was taken from [7], while the former was considered in [6].

When it comes to Monte Carlo simulations, we consider that systems start from an empty queue and use Matlab [16] to simulate M/M/1 queues starting with X0=0, without a warm-up period. The code used is an adaptation from the supplementary material of [7], available online. The number of replications/runs was increased from 500 (the number used by Chen and Zhou [7], for the CUSUM-P chart) to rep=5000, to obtain smoother ARL curves. In order to control the random number generation and thus save the results from each simulation, we used Matlab’s procedure rng. Each run is associated with fixed arrival and service rates and a maximum of N=6000 departures.

While comparing the CUSUM charts associated with complete and partial sampling schemes, we made sure that the design parameters verify ρ1=λ1μ1 and use the same simulation runs to compute estimates of the corresponding ARL.

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 Wn-chart whose expected number of departures until a false alarm is triggered equals 370.

The estimates of the RL percentage points, ARL, SDRL (standard deviation of the RL), CVRL (coefficient of variation of the RL) of the Wn-chart are in Table 2, whereas the corresponding values for the CUSUM-P and CUSUM-C charts can be found in Tables 3 and 4.

Table 2

Wn-chart for an M/M/1 system with ρ0=0.3,0.5: in-control and out-of-control (ρ=1.1ρ0,1.2ρ0,1.5ρ0) RL percentage points, ARL, SDRL and CVRL values – scenarios (i), (ii), (iii).

RL percentage points
ρ1%5%10%25%50%75%90%95%99%ARLSDRLCVRLScenario / UCL
0.36214110825751284911041696370.000367.9450.9944IC / 5.72998
0.334132564150298494643987215.823213.8280.9908(i)
0.3639164196191316410630138.272136.3410.9860
0.4524716377311915523752.79551.0490.9669
0.3351834902144267069171409307.732305.5120.9928(ii)
0.3651629761803585937701183258.783256.3980.9908
0.454112049114224371481738162.600159.7260.9823
0.3351529751793555887651175256.692254.5880.9918(iii)
0.364122255130258427555852186.694184.5500.9885
0.453711276212220026039988.28586.0610.9748
0.58244311025851184610991687370.000365.2600.9872IC / 7.74703
0.556142560139273450584896198.075193.5330.9771(i)
0.6410173885165271351537120.062115.7350.9640
0.753581531589412218543.07339.3940.9146
0.5571934831933816298171253276.084270.8760.9811(ii)
0.67162765148290477619949210.484204.8190.9731
0.75612173575143234302460105.32898.4300.9345
0.5561729711633215306881056233.068228.2040.9791(iii)
0.65132149111216356462707157.377152.4290.9686
0.75481123478914518728465.63160.6240.9237
Table 3

CUSUM-P chart for an M/M/1 system with ρ0=0.3,0.5: in-control and out-of-control (ρ=1.1ρ0,1.2ρ0,1.5ρ0) RL percentage points, ARL, SDRL and CVRL values – scenarios (i), (ii), (iii) and design parameter ρ1=1.1ρ0.

RL percentage points
ρ1%5%10%25%50%75%90%95%99%ARLSDRLCVRLScenario / UCL
0.329547614028050879610201572375.382330.0940.8794IC / 0.72
0.3324425693168291452574861219.666179.2220.8159(i)
0.3620324269117187286363551145.907110.4640.7571
0.4512202639588611914420267.18440.4620.6023
0.3325425795168292456577840219.875176.0630.8007(ii)
0.3620344368114186279356507142.217103.9340.7308
0.4513222739598712114620568.22640.2330.5897
0.3326425693166289452578830218.528175.4160.8027(iii)
0.3620364470116195288361539147.223109.6320.7447
0.4514212739588812214720768.35240.920.5987
0.527527614027550180110141518372.462325.1270.8729IC / 0.83
0.5523385085149261403524778196.523160.0640.8145(i)
0.61829386098160238302436122.82389.3500.7275
0.7512192333497210012117056.92733.1080.5816
0.5518334578143263418540802196.361168.8970.8601(ii)
0.61425335494159241309452120.77494.6290.7835
0.751016203045699812017553.77233.9960.6322
0.5513243569142271445584843200.622185.5410.9248(iii)
0.61019274791167262335532124.485110.7820.8899
0.757121525426710112718851.77938.2710.7391
Table 4

CUSUM-C chart: in-control and out-of-control (ρ=1.1ρ0,1.2ρ0,1.5ρ0) RL percentage points, ARL, SDRL and CVRL values for an M/M/1 system with ρ0=0.3,0.5 – design parameters: (λ1,μ1)=(λ0,μ0/1.1), scenario (i); (λ1,μ1)=(1.1λ0,μ0), scenario (ii); (λ1,μ1)=(1.1λ0,μ0/1.1), scenario (iii).

RL percentage points
ρ1%5%10%25%50%75%90%95%99%ARLSDRLCVRLScenario / UCL
ρ0=0.322466913026449679710301578368.209335.9700.9124IC / 1.26
0.3317293963108183276358526138.873108.0830.7783(i)
0.3613212640639814418026376.65052.8880.6900
0.457111420294054648632.13216.7240.5205
0.339628114127249679010161552370.602324.2750.8750IC / 1.27
0.3333445477122195284353518150.190101.8980.6785(ii)
0.36283642557610915218625788.97448.4470.5445
0.4521262934415163718944.03114.1110.3205
0.33355781392724957929841509368.550320.4740.8696IC / 0.96
0.3328435483140229344434628175.785130.5480.7427(iii)
0.362335436393148216267379115.35876.7530.6653
0.451522273649679010614254.74326.8430.4903
ρ0=0.52649721342674977869991601367.008330.0970.8994IC / 1.27
0.5517283864109181277362535139.107109.5010.7872(i)
0.6142127416410014217725577.26351.8060.6705
0.757121420294155658832.54417.0900.5252
0.54063831412755077959991563373.974325.1840.8695IC / 1.28
0.5530435377123199296366531153.551108.7130.7080(ii)
0.6243441547610915318826888.85850.7170.5708
0.7515222632415264728742.99515.2070.3537
0.532568013927148678610041546366.764324.4450.8846IC / 0.96
0.5525415385141238363450679181.432138.2090.7618(iii)
0.62133426094143210257375112.51773.5720.6539
0.751321253548678910514453.67927.3260.5091

Before discussing the results for each target value ρ0, we ought to mention that under all scenarios, the CUSUM-C chart seems to outperform the competing charts for most out-of-control values of the traffic intensity.

For ρ0=0.3,0.5 and under scenario (i), Tables 2 and 3 and also Figure 1 suggest that the Wn-chart has a better ARL performance than the CUSUM-P chart. Unlike the control statistic of the CUSUM-P chart,

Wn+1=max{0,Wn+Vn+1-Un+1}

comprises explicit information on the service time (Vn+1); thus, when the traffic intensity increases due solely to a decrease in the service rate, the Wn-chart is bound to trigger signals quicker than the CUSUM-P chart. Furthermore, bear in mind that the expected value and the variance of the waiting time (W) in the ergodic M/M/1 system are given by

E(W)=ρμ(1-ρ)andV(W)=ρ(2-ρ)[μ(1-ρ)]2

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 Wn are bound to be closer to the associated UCL and the Wn-chart is more prone to signal.

Under scenario (ii), the Wn-chart has clearly a feeble detection ARL performance, as depicted in Figure 1, for ρ0=0.3,0.5. This does not come as a surprise given the results previously reported in Section 2, in particular that the Xn-chart (resp. X^n-chart) proved to be faster than the Wn-chart in the detection of increases in ρ. On top of that, if ρ increases due to an upward shift in the arrival rate while the service rate remains constant, Wn is less likely to exceed the UCL than in scenario (i), therefore the larger out-of-control ARL and a smaller detection ability.

For ρ0=0.3 and under scenario (iii), Figure 1 leads us to conclude that the Wn-chart is the best option in the presence of large shifts in the traffic intensity; this is basically due to the typical initial inertia of CUSUM charts in such out-of-control situations.

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 ρ1, because the CUSUM-P statistic only depends on the design parameter and the traffic intensity itself, unlike the Wn-chart (resp. CUSUM-C chart) that depends on the values of the arrival and service rates (resp. design parameters λ1 and μ1).

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.

Figure 1
Figure 1

Log-ARL values of the CUSUM-C, CUSUM-P and Wn charts – M/M/1 system with ρ0=0.3 (left), 0.5 (right); scenarios (i) (top), (ii) (middle) and (iii) (bottom).

Citation: Stochastics and Quality Control 34, 1; 10.1515/eqc-2018-0026

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 Wn-chart; Monte Carlo simulation of the M/M/1 queue in order to estimate RL related measures of the CUSUM-P and CUSUM-C charts, proposed by Chen and Zhou [7].

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 λ1 and μ1 with respect to the in-control values λ0 and μ0 (see [7]). The design of the Wn-chart is far more straightforward because it does not depend on any design parameter, and therefore may be preferred in practice if we do not anticipate the out-of-control scenarios (ii) and (iii).

Curiously, a comparison between the performances of the CUSUM-P and Wn charts led us to believe that the CUSUM-P chart seems to be outperformed by the Wn-chart under scenario (i), i.e., when the traffic intensity increases due to a decrease in the service rate and the arrival rate remains unchanged. Moreover, the Wn-chart can be faster than the CUSUM-C chart in the presence of large shifts under two out-of-control scenarios.

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 Wn and proposed by Morais and Knoth [9]. Thus, a direction of future research comprises the derivation of ARL-unbiased versions of the CUSUM-C and CUSUM-P charts, such that: their in-control ARL take a pre-stipulated value ARL; the associated ARL curves attain a maximum when the traffic intensity is on target, thus it takes us less time (in average) to be alerted to any increase or decrease of the traffic intensity than to run into a false alarm.

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 M/M/1 queue, following the lines of what Chen and Zhou [7, Section 4] have done for the M/G/1 and M/M/s queues, and the subsequent a comparison of the performance of the Wn and these CUSUM charts.

Acknowledgements

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

References

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    U. N. Bhat, A control technique for the parameters of stochastic processes, Bull. Inst. Internat. Statist. 44 (1971), no. 2, 94–98.

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    U. N. Bhat, A statistical technique for the control of traffic intensity in Markovian queue, Ann. Oper. Res. 8 (1987), 151–164.

    • Crossref
    • Export Citation
  • [3]

    U. N. Bhat and I. Sahin, Transient behavior of queuing systems M/D/1, M/Ek/1, D/M/1 and Ek/M/1, Technical Memo M135, Department of Operations Research, Case Western University, Cleveland, 1969.

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    U. N. Bhat and S. Subba Rao, A statistical technique for the control of traffic intensity in the queuing systems M / G / 1 M/G/1 and G I / M / 1 GI/M/1, Operations Res. 20 (1972), 955–966.

    • Crossref
    • Export Citation
  • [5]

    D. Brook and D. A. Evans, An approach to the probability distribution of cusum run length, Biometrika 59 (1972), 539–549.

    • Crossref
    • Export Citation
  • [6]

    N. Chen, Y. Yuan and S. Zhou, Performance analysis of queue length monitoring of M/G/1 systems, Naval Res. Logist. 58 (2011), no. 8, 782–794.

    • Crossref
    • Export Citation
  • [7]

    N. Chen and S. Zhou, CUSUM statistical monitoring of M/M/1 queues and extensions, Technometrics 57 (2015), no. 2, 245–256.

    • Crossref
    • Export Citation
  • [8]

    S.-H. Kim, C. Alexopoulos, K.-L. Tsui and J. R. Wilson, A distribution-free tabular cusum chart for autocorrelated data, IIE Trans. 39 (2007), 317–330.

    • Crossref
    • Export Citation
  • [9]

    M. C. Morais and S. Knoth, On ARL-unbiased charts to monitor the traffic intensity of a single server queue, Proceedings of the XII. International Workshop on Intelligent Statistical Quality Control Hamburg 2016, Front. Stat. Qual. Control 12, Springer, Heidelberg (2018), 217–242.

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    M. C. Morais and A. Pacheco, On stochastic ordering and control charts for traffic intensity, Sequential Anal. 35 (2016), no. 4, 536–559.

    • Crossref
    • Export Citation
  • [11]

    A. M. Pacheco Pires, Sistemas M / M / r / n M/M/r/n: Comparação de sistemas com iguais taxas de chegadas e de serviço (M / M / r / n M/M/r/n systems: Comparing systems with the same arrival and service rates), Master’s thesis, Department of Mathematics, Instituto Superior Técnico, University of Lisbon, Lisbon, 1990.

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    A. G. Pakes, Queueing theory, Encyclopedia of Statistical Sciences. Volume 7, John Wiley & Sons, New York (1986), 483–489.

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    N. U. Prabhu, Foundations of queueing theory, Int. Ser. Oper. Res. Management Sci. 7, Kluwer Academic Publishers, Boston, 1997.

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    M. D. M. Santos, On control charts and the detection of increases in the traffic intensity of queueing systems, Master’s thesis, Department of Mathematics, Instituto Superior Técnico, University of Lisbon, Lisbon, 2016.

  • [15]

    M. Santos, M. C. Morais and A. Pacheco, Comparing short-memory charts to monitor the traffic intensity of single server queues, Stoch. Qual. Control 33 (2018), no. 1, 1–21.

    • Crossref
    • Export Citation
  • [16]

    MathWorks, Matlab, Version 9.0 (R2016a), 2016. Accessed from www.mathworks.com on 2016-06-08.

If the inline PDF is not rendering correctly, you can download the PDF file here.

  • [1]

    U. N. Bhat, A control technique for the parameters of stochastic processes, Bull. Inst. Internat. Statist. 44 (1971), no. 2, 94–98.

  • [2]

    U. N. Bhat, A statistical technique for the control of traffic intensity in Markovian queue, Ann. Oper. Res. 8 (1987), 151–164.

    • Crossref
    • Export Citation
  • [3]

    U. N. Bhat and I. Sahin, Transient behavior of queuing systems M/D/1, M/Ek/1, D/M/1 and Ek/M/1, Technical Memo M135, Department of Operations Research, Case Western University, Cleveland, 1969.

  • [4]

    U. N. Bhat and S. Subba Rao, A statistical technique for the control of traffic intensity in the queuing systems M / G / 1 M/G/1 and G I / M / 1 GI/M/1, Operations Res. 20 (1972), 955–966.

    • Crossref
    • Export Citation
  • [5]

    D. Brook and D. A. Evans, An approach to the probability distribution of cusum run length, Biometrika 59 (1972), 539–549.

    • Crossref
    • Export Citation
  • [6]

    N. Chen, Y. Yuan and S. Zhou, Performance analysis of queue length monitoring of M/G/1 systems, Naval Res. Logist. 58 (2011), no. 8, 782–794.

    • Crossref
    • Export Citation
  • [7]

    N. Chen and S. Zhou, CUSUM statistical monitoring of M/M/1 queues and extensions, Technometrics 57 (2015), no. 2, 245–256.

    • Crossref
    • Export Citation
  • [8]

    S.-H. Kim, C. Alexopoulos, K.-L. Tsui and J. R. Wilson, A distribution-free tabular cusum chart for autocorrelated data, IIE Trans. 39 (2007), 317–330.

    • Crossref
    • Export Citation
  • [9]

    M. C. Morais and S. Knoth, On ARL-unbiased charts to monitor the traffic intensity of a single server queue, Proceedings of the XII. International Workshop on Intelligent Statistical Quality Control Hamburg 2016, Front. Stat. Qual. Control 12, Springer, Heidelberg (2018), 217–242.

  • [10]

    M. C. Morais and A. Pacheco, On stochastic ordering and control charts for traffic intensity, Sequential Anal. 35 (2016), no. 4, 536–559.

    • Crossref
    • Export Citation
  • [11]

    A. M. Pacheco Pires, Sistemas M / M / r / n M/M/r/n: Comparação de sistemas com iguais taxas de chegadas e de serviço (M / M / r / n M/M/r/n systems: Comparing systems with the same arrival and service rates), Master’s thesis, Department of Mathematics, Instituto Superior Técnico, University of Lisbon, Lisbon, 1990.

  • [12]

    A. G. Pakes, Queueing theory, Encyclopedia of Statistical Sciences. Volume 7, John Wiley & Sons, New York (1986), 483–489.

  • [13]

    N. U. Prabhu, Foundations of queueing theory, Int. Ser. Oper. Res. Management Sci. 7, Kluwer Academic Publishers, Boston, 1997.

  • [14]

    M. D. M. Santos, On control charts and the detection of increases in the traffic intensity of queueing systems, Master’s thesis, Department of Mathematics, Instituto Superior Técnico, University of Lisbon, Lisbon, 2016.

  • [15]

    M. Santos, M. C. Morais and A. Pacheco, Comparing short-memory charts to monitor the traffic intensity of single server queues, Stoch. Qual. Control 33 (2018), no. 1, 1–21.

    • Crossref
    • Export Citation
  • [16]

    MathWorks, Matlab, Version 9.0 (R2016a), 2016. Accessed from www.mathworks.com on 2016-06-08.

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    Log-ARL values of the CUSUM-C, CUSUM-P and Wn charts – M/M/1 system with ρ0=0.3 (left), 0.5 (right); scenarios (i) (top), (ii) (middle) and (iii) (bottom).