Parameter and State Estimation in Queues and Related Stochastic Models: A Bibliography

P arameter and State Estimation in Queues and Related Sto c hastic Mo dels: A Bibliograp h y Azam Asanjarani ∗ , Y oni Nazarath y † . Abstract This is an annotate d bibliograph y on estimation and inference results for queues and related s to c hastic mo dels. The purp ose of this documen t is to collect and categorise w orks in the ﬁeld, allo wing for researc hers and practitioners to explore the v arious t yp es of results that exist. Our fo cus i s on pap ers that deal with mathematical queueing mo d els as w ell as r elated sto c h astic mo dels motiv ate d by queues . W e attempte d to mak e this bibliograph y exhaustiv e, yet there are p ossibly some pa- p ers that we ha ve missed. As it is up dated con tin uously , additions and commen ts are w elcomed. Note that this bibliograph y is also a companion to our surv ey of parameter and state estimation in queues [20]. Add itional wo rks not men tioned in this bibliograph y include the following related categories: (i) Methods for parameter estimation of p oint pro cesses (not inv olving queues) . (ii) Methods for paramet er estimation of sto cha stic matrix analytic models (not in volving queues). (iii) Optimal con trol of queues including bandit problems where state or paramete r estim ation is not directly considered. (iv) Inference, estimation and tomograph y of communicat ion netw orks not directly mo delled as queueing netw orks. (v) Analysis of queues with parameter uncertain ty , and robustness not directly inv olving inf erence. (vi) Road traﬃc modelli ng not i nv ol vi ng an explicit congestion or queueing mo del. ∗ The Univ er sity of Auc kla nd. † The Univ er sity of Queensland. 1 1 Chronological order with brief descriptions 1955 Co x [86]: An o v erview pap er of queueing theory outlining the philosoph y of estimating parameters of input pro cesses vs. p erformance pro cesses. 1957 Benes [40]: T ra nsien t M/M/ ∞ full observ ation o ve r a ﬁxed in terv al. Clark e [77]: M/M/1 MLE with full observ ation. The ﬁrst pap er. Sampling un til “busy time” r eaches a pre-assi gned v alue yields closed-form MLEs. 1961 Billingsley [48]: Bo ok on inference of Mark o v c hains. 1965 Co x [87]: An ov erview pap er on parameter estimation, separate analysis of input and service mec hanism a nd pro blems connected with the sampling of queueing pro cess. K ov alenk o [204]: On recov ering the c haracteristics of a system from observ a tions of the outgoing ﬂo w (in Russian) New ell [252]: A review of appro ximation metho ds for queues with a pplication to the ﬁxed-cycle traﬃc ligh t. W olﬀ [336]: Large sample theory for birth-death queues . 1966 Lilliefors [217]: Conﬁde nce in terv als for standard p erformance measuremen ts based on parameter error. 1967 Green b erg [139]: Diﬀeren t w ays of determining fo r ho w long to observ e a stationary M/M/1 queue (e.g. ﬁxed n um b er of arriv als, ﬁxed total observ a tion time, etc. 1968 Daley [92]: The serial correlation co eﬃcien ts of queue sizes in a statio nar y G I/M/1 queue are studied. Daley [93]: The serial correlation co eﬃcien ts of a (stationary) sequence of w aiting times in a stationary M/M/1, M/G/1 and G I/G/l queueing system are studied. 1970 Bro wn [55]: Estimating the G in M/G/ ∞ with arriv al and departure times without know - ing what customers they related to . 2 Ross [284]: Discusses iden tifying the distributions of GI/G/k uniquely based on observ a- tion of the queueing pro cess. 1971 P akes [256]: The serial correlation co eﬃcien ts of waiting times in the stationary GI/M/1 queue are studied. (completing [92] work) 1972 Go ya l and Harris [136]: MLE for queues with Poiss on arriv als with state-dep enden t general service times when queue size s are observ ed at departure p o in ts. Jenkins [179]: Compares the asymptotic v ariance of t w o es timators for M/M/1. Muddapur [245]: Ad ds a prior distribution to Clark e’s [77] approach . 1973 Harris [150]: An ov erview pap er presen ted the statistical analysis of queuein g systems with an emphasis on the estimation o f input and servic e parameters and/or distributions. Neal and Kucz ura [248 ]: Presen ts a ccurate appro ximation and asymptotic appro xima- tions (b y using renew al theory) for the v ariance of an y diﬀeren tiable functions of diﬀeren t traﬃc measuremen ts. Reynolds [278]: Bay esian approac h for estimation of birth death parameters. 1974 Aigner [4]: C ompares prop erties of v arious estimators for M/M/1 with cross-sectional data. Brillinger [53]: Estimates parameter for a linear time inv ariant mo del that generalizes the G/G/ ∞ queue. 1975 Keiding [190]: Analyses asymptotic prop erties of the MLE for a birth-and-death pro cess with linear rates (b oth birth and death). 1979 Thiagara jan and Harris [313]: Exp onen tial go o dness of ﬁt test for the se r vice times of M/G/1 based on observ ations of the queue lengths and/or w aiting t imes. 1980 Da v e and Shah [98]: MLE of an M/M/2 queue with heterogenous serv ers. Gordon and Dowdy [134]: In v estigates the eﬀect of parameter estimation errors on the p erformance in closed pro duct form queueing net w o rks. 3 1981 Basa w a and Pr abh u [37 ]: Estimates for non-parametric and parametric mo dels of single serv er queues o v er a horizon up to the n th departure epo c h. Also the m.l.e’s of the mean in ter-arriv al time and mean service time in an M/M/1 o bserv ed ov er a ﬁxed time-in terv al. W alrand [320]: Prop oses an elemen tary justiﬁcation of the ﬁltering formulas for a Marko v c hain and an analysis of the arriv al and departure pro cesses at a ./M/1 queue in a quasi- rev ersible netw ork. Grassmann [137]: This pap er sho ws that in the M/D/ ∞ queueing system with service time S , the optimal w a y to estimate the exp ected n umber in the system is b y sampling the system at time 0 , S, 2 S, · · · , k S . 1982 Halﬁn[144]: Finds the minim um-v ariance linear estimator fo r the expected v alue of a stationary sto c hastic pro cess o bserv ed ov er a ﬁnite time in terv al, whose co v ariance function is a sum of deca ying exp onen tials. Sc hrub en and Kulk arni [286]: Studies the in terface b et we en stochastic mo dels and ac- tual systems for the sp ecial case o f M/M/1 queue. 1983 Hernàndez-Lerma an d Marcus [157]: A daptiv e con tr o l of an M/G/1 queue ing system, where the con trol chooses the service rate to minimize costs. 1984 Baras, Dorsey , and Mak o wski [31]: Sta bility , parameter estimation and adaptiv e con t r o l for discrete-time comp eting queues. Esc henba ch [117]: Brieﬂy describ es metho ds and results in the statistical analysis of queueing system s. Edelman and McK ellar [113]: Commen ts on [98]. Hernàndez-Lerma and Marcus [158]: A daptiv e control o f priority assignmen t in a m ulti-class queue. Mac hihara [224]: The carried traﬃc estimate errors for dela y system mo dels are analyzed with emphasis on the analysis of the eﬀect of the holding time distribution on the estimate errors. W arﬁeld and F o ers [328]: A Bay esian metho d for analysing teletraﬃc measuremen t data is discuss ed. W o o dside, Stanford and Pagu rek [337]: Presen ts optimal mean square predic t o rs for queue lengths and dela ys in the stationary G I/M/m qu eue, based on a queue length mea- suremen t. 1985 Armero [14]: The p osterior distribution of traﬃc in tensit y a nd the p osterior predictiv e distribution of the w aiting time a nd n um b er of customers fo r a M/M/1/ ∞ F IFO queue are obtained giv en tw o indep enden t sample s of arriv al a nd service times. 4 W arﬁeld and F o ers [329 ]: Bay esian analysis for traﬃc in tensit y in M/M/c/K type mo d- els and in retrial mo dels. 1986 Subba R ao and H arishcha ndra [304]: Large normal approx imation test based for the traﬃc in tensit y parameter in GI/G/s queues. 1987 Bhat and Rao [46]: A ﬁrst ma jor surv ey on statistical analysis of queueing systems. Mcgrath, Gross and Sin gpurw alla [236]: A ttempts to illustrate Ba y esian approa ch through M/M/1 and M/M/1/K examples . McGrath and Singpurwa lla [237]: This is part II to [236] (without Gross). Here the fo cus is on in tegra t ing the “Shannon measure of information”(cross-en tropy) in the analysis. Ramalhoto [269]: Discusses estimation of generalizations of GI/ G / ∞ , i.e. random trans- lations whose distribution is par a meterized b y a certain function, h ( · ) . 1988 Basa w a and Prabh u [38]: Estimation of GI/G/1 with exp onen tial fa mily densities. F ull observ ation ov er [0 , T ] where T is a stopping time. Sev eral T ’s considered and asymptotic prop erties compared. Chen, Harrison, Mandelbaum, V an Ac k ere, W ein [64] : Empiric al ev aluation of a queueing net w ork mo del for semic onductor wafer f a brication. Harishc handra and Rao [149]: Inference for the M/ E k /1 queue. Jain and T empleton [174]: Estimation of GI/M/1 (and GI/M/1/m with m kno wn) parameters where the arriv al ra t e is either λ or λ 1 dep ending on the queue lev el. Nozari and Whitt [254]: Propose an indirect approac h to estimate a verage production in terv als (the length o f time b etw een starting a nd ﬁnishing work on eac h pro duct) using w ork-in-pro cess in v entory measuremen ts. 1989 F endic k and Whitt [120]: Prop oses measuremen ts and approximations to describe the v ariabilit y of oﬀered traﬃc to a queue (the v ariabilit y of the arriv al pro cess together with the service requireme n ts) and predicts the a v erage workload in the queue (whic h a ssumed to ha ve a single serv er, unlimited waiting space and a work -conserving service discipline). Glynn and Whitt [131]: Using the little’s L = λW and generalizations to infer L from W and the opp ositiv e. Han tler and Rosb er g [148]: P ar a meter estimation of M/M/c queue with parameters in sto c hastic v arying en vironmen t, ﬁrst doing the constan t inv aria nt deriv ation and then using in conjunction with Kalman ﬁlter for the time-v arying case. Jain and T empleton [175]: Sequen tial analys is view for M/ E k /1 queues . Sengupta [291]: Presen t an algorithm for compu ting the steady-state distribution of the w aiting time a nd queue length of the stable GI/ K / l queue. 5 1990 Ga v er and Jacobs [126]: T ransien t M/G/1 inference. Ga wlic k [127]: Applies the (QIE) to ethernet data. Larson [209]: This deals with “State Reconstruction” as opp osed to “ parameter inference” in what is called the “Queue Inference Engine” (QIE). This is the ﬁrst of man y pap ers on the idea of using transactional data to reconstruct an estimate of the queue length pro cess. Rubin and R obson [285]: Inference and estimation of n um b er of arriv als for a queueing system with losses due to bulking a nd a serv er that w orks a ﬁxed shift and sta ys to w ork after the shift. Small sample analysis as opp osed to asymptotic prop erties. 1991 Hall and Larson [146]: Mo diﬁes (extends) the QIE [209] to ﬁnite queues and to a case where there is data ab out exceedin g a certain lev el. Jain and T empleton [176]: Conﬁdence in terv als for estimation fo r M/M/2 with heteroge- nous serv ers. Jain [169]: Compares conﬁdence in terv als for ρ using sev eral methods and sampling regi- mens in M/ E k /1 queues . Larson [210]: An addendum to [209] reducing the computational complexit y f r o m O ( N 5 ) to O ( N 3 ) . Thiruv a iyaru, Basa w a and Bhat [315]: Large sample theory for MLEs of Jac kson net- w orks. 1992 Asm ussen [21]: pro v es that the stationary waiting time in a GI/PH/1 queue with phase- t yp e serv ice time is phase-t yp e. Asm ussen and Bladt [22]: The Matrix-exp onential distribution is in tro duced and some of its basic structural prop erties are giv en. F urther, an algo rithm for computing the waiting time distribution o f a queue with matrix-exponen tial servi ce times and general in ter-arriv al times is giv en. This algorithm is a sligh t g eneralization of the algorithm for computing the w aiting time distribution of GI /P H/ 1 queues. Basa w a and Bhat [33]: Presen ts sequen tial analysis methods f or the traﬃc in tensit y of single serv er queues. Bertsimas and Servi [41]: Impro v es on the O ( n 5 ) algorithm in [209] to O ( n 3 ) . Also presen ts an on-line a lg o rithm for estimating the queue length after each departure and includes time-v arying P oisson generalizations. Daley and Servi [94]: Con tinue s the track of the QIE, using tab o o probabilities. Heyde [159]: Quasi-like liho o d estimation metho ds for stationa ry pro cesses and que ueing examples. Jain [170]: Deriv es the relativ e eﬃciency of a parameter for the M/G/1 queueing sy stem based on reduced and full lik eliho o d functions. In addition, Mon te Carlo sim ulatio ns w ere carried out to study the ﬁnite sample prop erties for estimating the para meters of an M/G/1 queueing system . Kumar [208]: Studies the bias in the means of a ve rage idle time and a ve rage queue length estimates, o v er the in terv al [0, t], in a transien t M/M/1 queue. 6 Singpurw alla [296]: A discuss ion pap er ab out [314]. The same issue for QUEST A a lso has a rejoinder for the discuss ion. Thiruv a iyaru and Basa wa [314]: Discuss es empirical Bay es estimation for v ariations of M/M/1 queues and Jac kson net w orks. 1993 Daley and Servi [95]: Discuss a fairly general Mark ov c hain setting for describing a sto c hastic pro cess at intermed iate time p oin ts r in r ∈ (0 , n ) conditional on certain kno wn b eha viour of the pro cess b oth on the interv al a nd at the endpoints 0 and n . Glynn, Melamed and Whitt [129]: Constructs conﬁdenc e in terv als for estimators and p erform statistical tests b y establish ing a joint cen tral limit theorem fo r customer and time a ve rages b y applying a martingale cen tral limit theorem in a Mark ov fra mew ork. 1994 Armero and Ba yarri [17]: Presen ts a Ba y esian approach to predict sev eral quan tities in an M/M/1 queue in equilib rium. Armero and Bay arri, M.J. [16]: Bay esian “prediction” in M/M/1 queues is considered . The meaning is Bay esian inference f or steady state quan tities such as the distribution of queue lengths. Armero [15]: Another Bay esian inference pap er. Chandra and Lee [63]: Presen ts Ba y esian metho ds for inferring customer b eha vior f r o m transactional data in telecomm unications systems. Chen, W a lrand and Messersc hmitt [68]: P erhaps the ﬁrst "probing" pap er. Deals with arriv als in a deterministic service time queue and estimates the Pois son arriv al rates based on prob e dela ys. Jang and Liu [177]: Presen ts a new queueing formula applicable in man ufacturing whic h uses v ariables easier to estimate than the v ariance suc h as t he n umber of machin e idle p erio ds. Jones and Larson [182]: Dev elops an eﬃcien t algorithm for ev en t probabilities of order statistics and uses it for the queue inference engine ([209]). Pitts [263 ]: Analysis of non-parametric estimation o f the GI/G/1 queue input distribu- tions based on observ ation of the w aiting time. 1995 Duﬃeld, Lewis , O’Connell, Russell, and T o omey [108]: Estimates directly the ther- mo dynamic en tropy o f the data- stream a t an input-p ort. F rom this, the qualit y-of-service parameters can b e calculated rapidly . Jain [171]: Change p oint detection in an M/M/1 queue. Muth u and Sampa thkumar [246]: The maxim um lik eliho o d estimates of the parameters in v o lv ed in a ﬁnite capacit y priority queueing mo del are obtained. The precision of the maxim um lik eliho o d estimates is studied using lik eliho o d theory for Mark ov pro cesses. Masuda [234]: Prov ides suﬃcien t conditions under whic h the in tuition (based on partial observ ations) can b e justiﬁed, and inv estigates related prop erties of queueing systems. 7 1996 Basa w a, Bhat, and Lund [34]: MLE fo r GI/G /1 based on waiting time data. Dimitrijevic [102]: Considers the problem of inferring the queue length of an M/G/1 queue using transactional data o f a busy p erio d. Manjuna th and Molle [231]: In tro duces a new oﬀ- line estimation algorithm for the w aiting times of departing customers in an M/G/1 queue with FC F S service by decoupling the arriv al time constrain ts from the customer departure times. Massey , P ark er, and Whitt [233]: Estimate the par a meters o f a nonhomogeneous P oisson pro cess with linear rate ov er a ﬁnite in t erv al, based on the n umber of coun ts in measuremen t subin terv als. Sohn [298]: Simple M/M/1 Bay esian parameter estimation. Sohn [299]: Bay esian estimation of M/M/1 using sev eral competing metho ds. 1997 Armero and Ba ya rri [18]: Bay esian infere nce of M/M/ ∞ . Basa w a, Lund, and Bhat [36]: Extends [34] using estimating functions. Bhat, Miller and Rao [45]: A surv ey pap er, a decade after the previous Surve y by Bhat and Rao, [46]. Daley and Servi [96]: Computes t he distributions and momen ts of w aiting t imes o f customers within a busy perio d in an FCFS queuing sy stem with a P oisson arriv al pro cess b y exploiting an embedded Mark ov c hain. Glynn and T orres [130]: Deals with estimation of the tail prop erties of the workload pro cess in b oth the M/M/1 queue and queues with more complex arriv als suc h as MMPP . Ho and Cassandras [160]: A surv ey on perturbation theory . Pic k ands and Stine [261]: Discrete time M/G/ ∞ queue. T o yoizum i [316]: W aiting time inference in G/G /1 queues in a non-parametric manner using “Sengupta’s in v arian t relationsh ip”. 1998 Daley and Servi [97]: Computes t he distributions and momen ts of w aiting t imes o f customers within a busy perio d in an FCFS queuing sy stem with a P oisson arriv al pro cess b y exploiting an embedded Mark ov c hain. Ganesh, Green, O’Connell and Pit t s [124]: App ears lik e a “ visionary” pap er on the use of non-parametric Ba ye sian metho ds in net work managemen t. Insua, Wip er and Ruggeri [168]: Ba y esian inference for M/G/ 1 queues with either Erlang or h yp er-exp onen tial se rvice distributions. Mandelba um and Zelt yn [225]: Queuing inference estimation in netw orks. Ro drigues and Leite [281]: A Bay esian inference ab out the traﬃc in tensit y in an M/M/1 queue, without w orrying ab out n uisance parameters. Sharma and Mazumdar [293]: Prop o ses sev eral sc hemes that the call a cceptance con- troller, at the ente r ing no de o f an A TM net work, can use to estimate the traﬃc of the users on the v arious routes in the net w or k b y sending a probing stream. Wip er [335]: Perhaps complemen t s [168] with analysis of G/M/c queues with the G b eing Erlang or h yp er-exp onen tial rene w al processes. 8 1999 A chary a [3]: Analyses the rate of con vergen ce of the distribution of MLEs in G I/G/1 queues with assumptions on the distributions as b eing f ro m expo nen tial families. Con ti [82]: Ba ysian inference fo r a G eo/G/1 Discre te t ime queue. Bingham and P itts [49]: Non-parameteric estimation in M/G/ ∞ queues. Bingham and Pit t s [50]: Estimates t he arriv a l rate of an M/G/1 queue given observ a tions of the busy and idle perio ds of this queue. Jones [180]: Analyses queues in the prese nce of balking, using only the service start and stop data utilized in Larson’s Queue Inference Engine . Ro drigo and V azquez [280]: Analyses a general G /G/1 retrial queueing systems from a statistical viewpo int. Sharma [292]: Using the measuremen t to ols a v a ilable on the In t ernet, sugges ts and com- pares diﬀeren t estimators to estimate aggregate traﬃc in tensities at v arious no des in the net w o r k. 2000 Armero and Conesa [19]: Statistical analysis of bulk a r r iv al queues fro m a Bay esian p oin t of view. Duﬃeld [107]: Analyses the impact of measuremen t error within the framew or k of La r g e Deviation theory . Glynn and Zeevi [132]: Estimates tail probabilities in queues. Jain [172]: Sequen tial analysis. Jain and Rao [173]: In v estigates the problems of statistical inference for the GI/G/1 queueing system . Zheng and Seila [352]: Construct estimators for the limiting expected num b er of cus- tomers in the queue (and sev eral other p erformance measures) with b etter sampling prop- erties in comparison to the existing estimators. 2001 Alouf, Nain and T owsley [6]: Probing estimation for M/M/1/K queues using momen t- based estimators based on a v ariet y of computable p erformance measu res. Huang and Brill [163 ]: Deriving the minim um v ariance unbiase d estimator (MVUE) and the maxim um lik eliho o d estimator (MLE) of the statio nary probabilit y function of the n um b er of customers in a collection of indep enden t M/G/c/c subsystems. Jang, Suh and Liu [178]: Presen ts a new GI/G/ 2 queueing formula whic h uses a sligh tly diﬀeren t set of data easier to obtain than the v ariance of in ter- a rriv al time. P ascha lidis and V assilaras [259]: Buﬀer ov erﬂow probabilities in queues with correlated arriv al a nd service pro cesse s using larg e deviations. 2002 Con ti [83]: Non- par a metric statistical a nalysis of a discrete-time queueing sys tem is con- sidered and estimation of p erformance measures of the system is studied. Con ti and De Giov anni [85]: Considers p erformance ev aluation of a disc r ete-time G I/G/1 queueing mo del with a fo cus on the equilibrium distribution of t he waiting time. 9 Sohn [300]: Ev en though the title has “Robust” ’, this pap er app ears to b e a standard M/M/1 Ba ysian infere nce pap er using the input data. Zhang, Xia, Squillan te and Mills [345]: A general approac h to infer the p er-class ser- vice times at diﬀeren t serv ers in multi-clas s queueing mo dels. 2003 Cao, A ndersson, Nyb erg, and Kihl [58]: W eb serv er p erformance is mo deled via an M/G/1/K*PS queue for whic h the authors a lso carry out maxim um lik eliho o d estimation of the parameters. Pic hitlamk en, Deslauriers, L’Ecuyer, and A vramidis [260]: This is a sim ulatio n mo delling pap er whe re the authors also carry out parameter estimation for the sim ulation mo del data via a real data set. 2004 Ausín, Wip er and Lillo [24]: Ba y esian inference of M/G / 1 using phase t yp e represen- tations of the G. Con ti [84]: A Ba yes ian non-parametric approac h to the analysis of discrete-time queuei ng mo dels. F earnhead [119]: Using forw ard- bac kw ar d algorithm to do inference for M/G / 1 and Er/G/1 queues. Hall and Park [145]: An M/G/ ∞ non-parametric pap er. W ang, Chen, and Ke [192]: Maxim um lik eliho o d estimates and conﬁde nce in terv a ls o f an M/M/R/N queue with ba lking and heterogeneous serv ers. 2005 Bladt and Sørensen [51]: Lik eliho o d inference for discretely observ ed Mark ov jump pro cesses with ﬁnite state space. Bro wn, Gans, Mandelbaum, Sak o v, Shen, Zelt yn a nd Z hao [54]: Ma jor pap er dealing with telephone call cen tre data analysis. Hei, B ensaou and T sang [155]: Probing fo cusing on the inte r-departure SCV of the probing stream in tandem ﬁnite buﬀer queues. Mandjes and v an de Meen t [227]: Prop ose an approa ch to accurately determine the burstiness of a net w o rk link on small time-scales (for instance 10 ms), by sampling the buﬀer o ccupancy (for instance) ev ery second. Prieger [266]: Sho ws that the MLE based on the complete inter-arriv al and service times (IST) dominates the MLE based on the n um b er of units in servic e (NIS), in terms of ease of implemen tatio n, bias, and v ariance. Ross and Shan t hikumar [282]: Estimating eﬀectiv e capacit y in Erlang loss systems under comp etition. Neuts [251]: Reﬂections on statistical metho ds for complex sto c hastic systems. 2006 Castellanos, Morales, Ma yoral , F ried and Armero [62]: Dev elops a Ba y esian analysis of queueing systems in applications of the mac hine in terference problem, lik e job-shop type systems , telecomm unication traﬃc, semiconductor man ufacturing or transp ort. 10 Chic k [70]: A surv ey c hapter on sub jectiv e probabilit y and the Ba y esian approac h, specif- ically in Mon te-Carlo sim ulation, y et giv es some insigh t in to que ueing inference. Ch u and Ke [73]: Cons truction of conﬁdence interv als of mean resp onse time fo r an M/G/1 F CFS queueing system. Doucet, Mon tesano Jasra [106]: Presen ts a trans-dime nsional Seque ntial Mon te Carlo metho d for online Ba ye sian inference in partially observ ed p oin t pro cesse s. Hansen and Pitts [147]: Non-parametric estimation of the servic e time distribution and the traﬃc in tensit y in M/G/1 queues based on observ a tions of the w orkload. Hei, Bensaou and T sang [156]: Similar to [155] but here the fo cus is on inter-departure SCV of the t wo consecutiv e probing pac k ets. Ke and Ch u [186]: Prop oses a consisten t and asymptotically normal estimator of intens ity for a queueing system with distribution-free in ter- arriv al and service times. Kim, Shira vi, and Min [203]: This pap er deals with net work congestion ana lysis via appro ximating pro cesses and uses real netw ork trace data for parameter ﬁtting o f self-similar net w o r k traﬃc using the index of disp ersion fo r coun ts and co eﬃcien t of determination. Liu, Heo, Sha and Zh u [221]: Prop oses a queueing-mo del-based a daptiv e con tro l ap- proac h for controlling the p erfor mance of computing systems. Liu, W ynter, Xia and Zhang [222]: presen ts an a pproac h for solving the problem of calibration of mo del parameters in the queueing netw or k framew ork using inference tec h- niques. Ro drigo [279]: Analyse the M/G/1 retrial queue from a statistical viewp oint. W ang, Ke, W ang and Ho [325]: Studies MLE and conﬁdence in terv als of an M/M/R queue with heterogeneous serv ers under steady-state conditions. 2007 Ausín, Lillo and Wip er [23]: Considers t he problem of designing a GI/M/c queueing system. Ch u and Ke [74]: Prop oses a consisten t and asymptotically nor mal estimator o f the mean resp onse time for a G/M/1 queueing system, whic h is based on the empirical Laplace function. Ch u and Ke [75]: Estimation a nd conﬁdence in terv al of mean resp onse time fo r a G /G/1 queueing system using data-based recursion relation a nd b o ot strap metho ds. Morales, Castellanos, Ma yoral, F ried and Ar mero [243]: Exploits Ba y esian criteria for designing an M/M/c//r queuei ng system with spares. P ark [257]: The use of auxiliary functions in non-parametric inference f or the M/G/ ∞ queueing mo del is considered. Ross, T aimre and Poll ett [283 ]: Estimation of rates in M/M/c queues using observ ations at discrete queues and MLE estimates o f an approx imate Orenstein Ullen b ec k (OU) pro cess. 2008 Ausín, Wip er and Lillo [25]: Bay esian inference for the transien t b eha viour and duration of a busy p erio d in a single serv er queueing system with general, unkno wn distributions for the in ter-arriv al and service times is in v estigated. Basa w a, Bhat and Zhou [35]: P arameter estimation based on the diﬀerences of tw o p ositiv e ex p onen tial family random v ariables is studied. 11 Casale, Cremonesi and T urrin [60]: Prop osed service time estimation tec hniques based on robust and constrained optimization. Casale, Zhan g and Smirni [61]: Sev eral con tr ibutions to the Mark ovian traﬃc analysis. Choudh ury and Borthakur [71]: Bay esian-based tec hniques for analysis of the M/M/1 queueing mo del. Dey [100]: Ba y es’ estimators of the traﬃc in tensity and v arious queue c haracteristics in an M/M/1 queue under quadratic error loss function hav e b een deriv ed. Ke, Ko and Sheu [189]: Prop oses an estimator for the exp ected busy p erio d of a con- trollable M/G/1 queue ing system in whic h the serv er applies a bicriterion p olicy during his idle p erio d. Ke, Ko and Chiou [188]: Presen ts a sensitivit y inv estigation of the exp ected busy p erio d for a controllable M/G/1 queueing system b y means of a fa ctorial design statistical analysis. Kim and Park [202]: In tro duces metho ds of que ue inference whic h can ﬁnd the in ternal b eha viours of queueing systems with only external observ ations, arriv al and departure time. Sutton and Jordan [307]: Analysing queueing net works from the probabilistic mo delling p ersp ectiv e, applying inference metho ds from graphical mo dels that aﬀor d signiﬁcan tly more mo delling ﬂexibilit y . Ramirez, Lillo and Wip er [270]: Considers a mixture of t wo-parameter Pareto distribu- tions as a mo del for heavy - tailed data and use a Ba yes ian a ppro ac h based on the birth-death Mark ov chain Mon te Carlo algorithm to ﬁt this mo del. 2009 Baccelli, Kauﬀmann and V eitc h [27]: P oin ts out the imp ortance of inv erse pro blems in queueing theory , whic h aim to deduce unkno wn parameters of the system based on partially observ ed tr a jectories. Baccelli, Kauﬀmann and V eitch [28]: Ev aluates the algorithm prop osed in [27]. Comert and Cetin [81] Presen ts a real-time estimation of queue lengths from the lo cation information of prob e v ehicles in a queue at an isolated and under-saturated in tersection. Ch u and Ke [76]: Constructs conﬁdence in terv als of in tensit y for a queueing system, whic h are based on four diﬀeren t bo otstrap metho ds. Duﬀy and Meyn [110]: Conjectures and presen ts support for this: a consisten t sequence of non-parametric estimators can b e constructed that satisﬁes a large deviation princ iple. Gorst-Rasm ussen Hansen [135]: Prop oses a framew or k based on empirical pro cess tec h- niques for inference ab out w aiting time and patience distributions in m ulti-serv er queues with abandonmen t. Hec kmull er and W olﬁnger [153]: Prop oses metho ds to estimate the parameters of arriv al pro cesses to G /D/1 queueing systems only based on observ ed departures from the system. Ibrahim and Whitt [165]: Studies the p erformance o f alternativ e real-time dela y esti- mators based on recen t customer delay exp eriences. Kiessler and Lund [193]: A note that considers traﬃc inten sit y estimation in the classical M/G/1 queue. Ke and Ch u [187]: Prop oses a consisten t and asymptotically normal estimator of intens ity for a queueing system with distribution-free in ter- arriv al and service times. Kraft, Pa checo-San c hez, Casale and Da wson [205]: Prop oses a linear regression metho d and a maxim um lik eliho o d tec hnique fo r estimating the service demands of requests 12 based on the measurem en t of their response times instead of their CPU utilization. Liu, W u, Ma, and Hu [220]: Presen ts an approac h to estimate time-dep enden t queue length ev en wh en the signal links are congested. Mandjes and Żuraniewski [229 ]: Dev elops que ueing-based pro cedures to (statistically) detect o verload in comm unication net work s, in a setting in whic h each connection consumes roughly the same amoun t of bandwidth. Mandjes and v an De Meen t [228]: The fo cus is on dimensioning as the approach for deliv ering performance requiremen ts of the net work. Nam, K im and Sung [247]: Estimates the a v ailable bandwidth for an M/G/ 1 queueing system. No v ak and W atson [253]: Presen ts a tec hnique to estimate the arriv a l rate from dela y measuremen ts, acquired using single-pac k et probing. 2010 Chen and Zhou [66]: Prop oses a non-linear quan tile regression mo del f or the relationship b et wee n stationary cycle time quantile s and corresp onding throughput r a tes of a man ufa c- turing system. Duﬀy and Meyn [109]: Deals with larg e deviations sho wing that in broad g enerality , that estimates of the steady-state mean p osition of a reﬂected random walk hav e a high lik eliho o d of o ver-estim ation. F rey and Kaplan [122]: In tro duces an algorithm for queue inference problems in v o lving p erio dic rep orting dat a . Pin, V eitch, and Kauﬀmann [262]: This pap er incorp orates queueing theory results in the application of netw ork tomograph y where statistical estimation of dela ys in a m ulticast tree using an EM algorithm is dev elop ed. Gans, Liu, Mandelbau m, Shen, and Y e [125]: Studies op erational heterogeneit y of call cen ter agen t s where the proxy for heterogeneit y is agen ts’ service times (call durations). Hec kmuel ler and W olﬁnger [152]: Prop oses metho ds to estimate the parameters of arriv al pro cesses to G /D/1 queueing systems only based on observ ed departures from the system. Pin, V eitc h and Kauﬀmann [262]: F o cuses on a sp eciﬁc delay tomogra phic problem o n a mul ticast diﬀusion t r ee, where end-to-end dela ys are observ ed at ev ery leaf of the tree, and mean so journ times are estimated for ev ery no de in the tree. Ramirez-Cob o, Lillo, Wilson and Wip er [272]: Presen ts a metho d for carrying out Ba ye sian estimation for the double P areto log normal distribution. Sutton and Jordan [308]: Presen ts a viewp oin t tha t com bines queueing net work s and graphical mo dels, allo wing Marko v c hain Mon te Carlo to b e applied. Xu, Zhang and Ding [338]: Discusse s testing h yp o t heses and conﬁdence regions with correct lev els for the mean so journ time of an M/M/1 queueing system. Zhang and Xu [346 ]: Discuss constructing conﬁd ence in terv als o f p erformance measures for an M/G/1 queueing system. Zuraniewski, Mandjes and Mellia [354]: Explores tec hniques for detecting unan tici- pated load c hanges with a fo cus on large-deviations based tec hniques dev elop ed earlier in [229]. 13 2011 Abramo v [1]: Statistic al b ounds for certain o utput c haracteristics of the M/GI/1/n and GI/M/1/n loss queueing systems are deriv ed on the basis of large samples of an input c haracteristic of these systems . Amani, Kihl and Rob ertsson [7]: An applications pap er to computer syste ms. Ban, Hao, and Sun [30]: Studies ho w to estimate real time queue lengths a t signaliz ed in tersections using the in tersection trav el times collected from mobile traﬃc sensors. Chen, Nan, Zhou [65]: In v estigates the statistical pro cess control application for moni- toring queue length data in M/G/1 systems. F eng, Dub e, and Zhang [121]: Considers estimation problems in G/G/ ∞ queue under incomplete information. Sp eciﬁcally , where it is infeasible to track eac h individual job in the system and only a ggregate statistics are kno wn or observ able. V eeger, Etman, Lefeb er, A dan, V an Herk, and Ro o da [318]: Predicting cycle time distributions for in tegrated pro cessing works tations: an agg regate mo delling approac h. Grüb el and W egener [141]: In M/G/ ∞ sys tems, considers the matc hing and exp onen- tialit y problems where the only observ atio ns are the order statistics asso ciated with the departure times and the order in whic h the customers arriv e and depart, respectiv ely . Ibrahim and Whitt [166]: Dev elops real-time dela y predictors for many - serv er service systems with a time-v arying arriv al rate, a time-v arying n umber of serv ers, a nd customer abandonmen t. Mandjes and Zuraniewski [230]: M/G/ ∞ c hange p oin t detection using la r g e deviations. Manoharan and Jose [232]: Considers an M/M/1 queueing system with customer im- patience in the form of random balking. McCab e, Martin and Harris [235]: Presen ts an eﬃcien t probabilistic forecast of in teger- v alued random v ariables that can b e interprete d as a queue, stock, birth and death pro cess or branc hing pro cess. P ark, Kim and Willemain [258]: Prop oses new approac hes that can a nalyse the unob- serv able queues using external observ ations. Sousa-Vieira [301]: Considers the suitabilit y of the M/G/ ∞ pro cess fo r mo delling the spatial and qualit y scalability extensions of the H.264 standard in video traﬃc mo delling. Sriniv as, Rao and Kal e [303]: Maxim um lik eliho o d and uniform minim um v ariance un biased es timators of measures in the M/M/1 queue are obtained and compared. Sutton and Jordan [309]: A Bay esian inference pap er b y computer system s researc hers. 2012 Duﬀy and Meyn [111]: Large deviation asymptotics for busy p erio ds for a queue. F abris-Rotelli, Kr aam wink el, and et al [118]: A historical and theoretical o v erview of G/M and M/G queue ing pro cesse s. Hu and Lee [162]: Conside r a parameter estimation problem when the state pro cess is a reﬂected f ractional Bro wnian motion (RF BM) with a non-zero drift parameter and t he observ ation is the asso ciated lo cal t ime pro cess. Jones [181]: Remarks on queue infere nce from departure da ta a lo ne and the imp orta nce of the queue inference engine. Kauﬀmann [185]: Prop oses a new approach, on the basis of existing TCP connections and reac hing therefore a zero probing o ve rhead based on the theory of inv erse problems in bandwidth sharing net w o r ks. 14 Kim and W hitt [195]: Statistical analysis with Little’s la w. Kim and W hitt [194]: Contains supplemen tary martial to [195]. Kim and W hitt [197]: Estimating w a iting times with the time-v arying Little’s la w. Kim and W hitt [196]: App endix to [197]. McVinish and Pol lett [238]: Uses estimating equations to get estimators for M/M/c queues and related mo dels. P erformance is compared to [283]. Mohamma di and Salehi-Rad [242]: Exploits the Ba y esian inference and prediction for an M/G/1 queuing mo del with optional second r e- service. Nelgabats, Nov and W eiss [249]: M/G/ ∞ estimation. Ren and Li [277]: Ba y esian estimator of t he traﬃc in tensit y in an M/M/1 queue is deriv ed under a new w eigh ted square error loss function. Ren and W ang [276]: Similer to [2 77]. Ba yes ian es t imato rs o f the traﬃc intens ity in an M/M/1 queue are deriv ed under a precautionary loss function. Whitt [331]: F itt ing birth-and-death queueing mo del to data. 2013 Larson [211]: A brief review on QIE. A chary a, Ro dríguez-Sánc hez and Villarreal-Ro dríguez [2]: Presen ts the deriv ation of maximum lik eliho o d estimates for the arr iv al rate and service rates in a stationary M/M/c queue with heterogeneous serv ers. Cho w [72]: Analysis of queueing mo del based on c haotic mapping. Li, Chen, Li and Zhang [216]: Prop oses a new algorithm based on the temp oral–spatial queueing mo del to describe the f ast tr av el-time v ariations using only the sp eed and headw ay time series that is measured at upstream a nd downstre am dete ctors. Wiler, Bolandifar, Griﬀey , P oirier, and Olsen [334]: This pap er applies queueing theory and inference of mo del para meters to deriv e and v alidate a nov el queuing theory- based mo del that predicts the eﬀect of v arious patien t cro wding scenarios on patien ts. W eerasinghe and Mandelb aum [330]: studies the tradeoﬀ b et wee n blo c king a nd aban- donmen t, with cost accum ulated ov er a random, ﬁnite time horizon of a con trolled queueing system of the G/M/n/B+M t yp e with many serv ers and impatien t customers . 2014 An tunes, Jacinto and Pac heco [10]: estimation of the arriv al rate and the service time momen ts of an M/G/1 queue with probing, i.e., with sp ecial customers (prob es) enterin g the system. The prob e inte r-arriv a l times are i.i.d. and prob e service times follo w a general p ositiv e distribution. The only observ ations used are the arriv al times, service t imes a nd departure times of prob es. W e deriv e the main equations from whic h the quan tities of in terest can b e estimated. T w o particular pro b e arr iv als, deterministic and P o isson, are in v estigated. Azriel, F eigin and Mandelbaum [26]: Prop oses a new mo del called Erlang- S, where “S” stands for Serv ers where there is a p o ol of presen t serv ers, some of whom are a v ailable to serv e customers from the queue while others are not, and the pro cess of b ecoming a v ailable or una v ailable is mo delled explicitly . Dinha, Andrewa and Nazarath y [103]: A conceptual and n umerical contribution to design and con trol of sp eed-scaled systems in view of parameter uncertain t y . 15 Edelmann and Wichelh aus [114]: T wo diﬀeren t nonparametric estimation approach for discrete-time sto c hastic netw orks of G eom X /G/ ∞ queues where the observ ation consists of the external arriv al and external departure pro cesses at the no des o ve r some time. He, Li, Huang and Lei [151]: Considers the queuing system a s a blac k box a nd derive a p erformance index for the qu euing system b y the principle of maximum en tro p y only on the assumption that the queue is stable. Kannan and Jabarali [183]: This pap er deals with maxim um lik eliho o d estimation pa- rameters for a v ariant of an M/M/1 queue with v acations. Kim and Whitt [199]: Considers diﬀeren t issues in testing the suitabilit y of the nonho- mogeneous P oisson pro cess as an arriv al pro cess with service sys tem data. Kim and Whitt [198]: (in t he follo wing of [19 9 ]) sho ws that call cen ter and hospital arriv als are we ll mo delled b y nonhomogeneous P oisson pro cesses . Sendero vic h, W eidlich , Gal, and Mandelbaum [288]: Establish a queueing p ersp ec- tiv e in op erational pro cess mining and demonstrate the v alue of queue mining using the sp eciﬁc op eratio nal problem of online dela y prediction. Y om-T o v and Mandelbaum [339 ]: Analyse s a queueing mo del, where customers can return to service sev eral times during their so j ourn within the sys tem. 2015 Bakholdina1 and Gortsev [29]: F o cused o n the problem of optimal estimation of the states of the mo dulated semi-sync hronous in tegrated ﬂow o f ev en ts. Burk ato vsk a ya , Kabano v a, and V orob eychik ov [319]: CUSUM algorithms for pa- rameter estimation in queueing systems where the arriv al pro cess is a Marko v- mo dulated P oisson pro cess. Caho y , Pol ito, and Phoha [57]: Statistical analysis of fractional M/M/1 queue and fra c- tional birth-death pro cesses; the p oin t pro cesse s go vern ed b y diﬀerence diﬀeren tial equations con taining fractional deriv ativ e operato rs. Chen and Zhou [67]: Prop ose the cumulativ e sum (CUSUM) sc hemes to eﬃcien tly mon- itor the p erformance of t ypical queueing systems based on diﬀeren t sampling sc hemes. Dong and Whitt [104 ]: Explores a sto c hastic grey-b ox mo delling of queueing systems by ﬁtting birth-and-death pro cesses to data. Dong and Whitt [105]: Using a birth-a nd- death pro cess to estimate the steady-State distribution of a p erio dic queue. Efrosinin, Winkler, and Martin [115]: Considers the problem of estimation and con- ﬁdence in terv al construction of a Mark ov ia n con tr o llable queuein g system with unreliable serv er and constan t retrial p o licy . Goldenshluger [133]: Non- parametric estimation of service time distribution of the M/G/1 queue from incomplete data on the queue. Gurvic h, Huang and Mandelbaum [143]: Prop oses a diﬀusion appro ximation for a man y-serv er Erlang-A queue . Kim and W hitt [200]: Similar to [199] Liu, W u, and Mic halop oulos [219]: Impro ve s queue size estimation b y prop osing dif- feren t ramp queue estimation algorithms. Moha jerzadeh, Y aghmae e, and Zahmatk esh [241]: Prop osed a metho d to prolong the net w o r k lifetime and to estimate t he target parameter eﬃcien tly in wireless sensor net works . 16 Sc hw eer and Wichelh aus [287] Estimation of the service time distribution in the discrete- time G I/G / ∞ -queue based solely on information on the arriv al and departure pro cesses is considered. The fo cus is put on the estimation approac h via the so called“sequenc e of diﬀerence” a nd pro ving a functional cen tral limit theorem for the resultan t estimator. Sendero vic h, Leemans, Harel, Gal, Mandelba um, and v an der Aalst [289]: Ex - plores the inﬂuence of a v ailable information in the log on the a ccuracy o f the queue mining tec hniques. Sendero vic h, W eidlic h, Gal, Mandelba um [290]: Queue mining for dela y prediction in m ulti-class service pro cesses . Sriniv as and Kale [302]: Compares the Maxim um Lik eliho o d (ML) and Uniformly Min- im um V ariance Un biased (UMVU) estimation for the M/D/1 queueing systems . Sutartoa and Jo elian to [305]: Presen ts an o v erview of urban traﬃc ﬂow from the per- sp ectiv e of system theory and sto c hastic con trol. W ang and Casale [326]: Prop oses maxim um lik eliho o d (ML) estimators for servi ce de- mands in closed queuei ng net w orks with load- indep enden t and load- dep enden t stations. W ang, Pérez, and Casale [327]: A soft ware for parameter estimation. Whitt [332]: Sequel to [104] and [106]. Establishes man y-serv er heav y-traﬃc ﬂuid limits f o r the steady-state distribution and the ﬁtted birth and death rates in perio dic Mt/GI/ inf ty mo dels. Zhan, Li, and Ukkusuri [343]: In the contex t of transp ortation engineering, with the complete arriv al and departure information, a car-follo wing based sim ulation sch eme is ap- plied to estimate the real-time queue length for eac h lane. 2016 Amini, Ped arsani, Sk abardonis, and V araiy a [8 ]: Queue-length estimation using real- time traﬃc data. An tunes, Jacin t o, Pac heco, and Wichelha us [12]: Uses a probing strategy to esti- mate the time dependen t traﬃc in tensit y in an M t /G t /1 queue, whe re the arr iv al rate a nd the general service-time distribution c hange from one time in terv a l to another, and deriv e statistical prop erties of the prop osed estimator. An usha, Sh arma, V ana jakshi, Subramanian, and Rilett [13]: Dev elops a mo del- based sc heme t o estimate the n umber o f v ehicles in queue and the total dela y . Comert [78]: This pap er, motiv ated b y the ﬁeld of traﬃc engineering, dev elops estimators for mark et p enetration lev el and a r r iv al rates based on queue lengths from prob e v ehicles at isolated traﬃc in tersections. Cruz, Quinino and Ho [89]: Uses a Ba yes ia n techn ique, the sampling/imp ortance re- sampling metho d to estimate the parameters o f multi-se rver queueing systems in whic h in ter-arriv al and service times are expo nentially distribu ted. Ghorbani-Ma ndolak ani and Salehi Rad [128]: Deriv es the ML and Ba y es estimators of traﬃc in tensit y and asymptotic conﬁdence in terv als for mean sys tem size of a tw o-phase tandem queueing mo del with a second optional servic e and random feedbac k and tw o het- erogeneous serv ers. Krishnasam y , Sen, Johari, and Shakk ot tai [206]: Considers regret analysis of a serv er allo cation problem where service rat es of serv ers are unkno wn. This analysis is in the con text of m ulti-armed bandits. 17 Morozo v, Nekraso v a, Pe shk o v a, and Rum y an tsev [244]: Dev elops a nov el approac h to conﬁdence estimation of the stationa r y measures in high p erformance m ulti-serv er queue- ing systems . Quinino an d Cr uz [267]: Describ es a Bay esian metho d for sample size determination fo r traﬃc in tensit y estimation. Zammit, F abri a nd Scerri [341]: A self-estimation algorithm is presen ted to jointly estimate the states and mo del parameters. Zhang, Xu, and Mi [347]: Considers the h yp othesis tests of p erformance measures for an M/E k /1 queueing syste m. 2017 Cruz, Quinino, and H o [90]: Ba y esian estimation of traﬃc inte nsit y based o n queue length in a m ulti-serv er M / M /s queue. den Bo er and Mandjes [99]: Considers con vergenc e rates of Laplace-tra nsform based estimators. Gu, Qian, a nd Zhang [142 ]: A Ba yes ia n probabilistic mo del alo ng with an expecta- tion–maximization extended Kalman ﬁlter (EM-EKF) algorithm is prop osed for traﬃc state estimation of urban road net w orks usin g a link queue mo del. Kim, Whitt, and Cha [201]: A data-driv en model of an app oin tmen t- generated arriv al pro cess at an outpatien t clinic. Li, T ang, Y ao, and Li [215]: Prop oses a cycle-b y-cycle queue length estimation metho d using only prob e data without the foregoing assump tion for signalized in tersections. Quinino and C r uz [268]: A Ba y esian me tho d is describ ed for sample size determination for traﬃc in tensit y estimation o f an M/M/1 queue. Sutarto, Jo elian to, and N ugroho [306]: Dev eloping a sto chastic mo del of queue length at a signalized in tersection. Whitt and Zhang [333]: Dev elops an aggregate sto chastic mo del of an emergency de- partmen t based on a careful study of data on individual patien t arriv al times a nd length of sta y . Zammit, F ab ri, an d Scerri [342]: Online state and multid imensional parameter estima- tion for a macrosc opic mo del of a traﬃc junction based on the Exp ectation-Maximization algorithm and m ultidimensional Robbins-Monro sto c hastic approxi mation. 2018 An, W ub, Xiaa, and Huanga [9]: This pap er from the ﬁeld of tra ﬃc engineering fo cuses on real-time queue length estimation in the con text of signalized in tersections. Almeida and Cruz [5]: The Jeﬀreys prior is prop osed to obtain the p osterior and predic- tiv e distributions of some para meters o f in terest. Samples are obtained through sim ulation and some p erformance c haracteristics are ana lyzed. Cruz, Almeida, D’Angelo, and v an W o ensel [88]: In v estigating the ﬁnite-sample b eha viour of some w ell-kno wn metho ds for the estimation of single-serv er ﬁnite Mark ovian queues or, in Kendall notat io n, queues, namely , the maxim um like liho o d estimator, Bay esian metho ds, and b o otstrap corrections. Oza wa [255]: Analysis of the stabilit y condition of a tw o- dimensional QBD pro cess and its application to ev aluate the eﬃciency of t w o-queue mo dels. 18 P olson and Sok olo v [264]: Dev elops an eﬃcien t par ticle learning algorithm for real time online inference of states and parameters. This requires a tw o-step approac h, ﬁrst resampling the curren t particles with a mixture predictiv e distribution and second propagation of states using the conditional p o sterior distribution. Suy ama, Qu inino, and Cruz [310]: Estimators fo r the parameters o f the Mark ovian m ultiserv er queues are presen ted, from samples that are the num b er of clien ts in the system at arbitrary p oin ts and their so jo urn t imes. 2019 Bhat and Basa wa[44]: An ov erview of the literature on the use of the maxim um lik eliho o d metho d for estimating parameters in queueing mo dels. Emami, Sa rvi, and A sadi Baglo ee [116]: A neural net w ork algorithm fo r queue length estimation based on the concept o f k-leader connecte d vehic les. Li, Ok amura , and Dohi [212]: supp ose an Mt/M/1/K queuein g system whose job a r riv al follo ws a Non-homog eneous P oisson Pro cess (NHPP) and prop ose a parameter estimation metho d for the NHPP approx imately from the utilization data based on the maximum lik eliho o d estimation (MLE) via the expectation maximization (EM) algorithm. Mei, Gu, Ch ung, Li, and T ang [239]: A Ba ye sian approac h for estimating ve hicle queue lengths at signalized in tersections using prob e veh icle data. Ra vner, Bo xma, and Mandjes [274]: Dev elops estimation sc hemes for a Lévy-driv en queue b y sampling the w o rkload pro cess at P oisson times. T an, Y ao, T ang, and Sun [312]: Cycle-based queue length estimation b y f using real- time and historical prob e vehi cle tra jectory data, through a statistical parameter estimation metho d, i.e., maxim um lik eliho o d estimation (MLE) Zhao, Zheng, W ong, W ang, Meng, and Liu [351]: V a r io us metho ds fo r queue length and traﬃc v olume estimation using prob e ve hicle tra jectories. 2020 Cruz, Almeida, D’Angelo, and v an W o ensel [88]: A bias-corrected ve r sion of MLE estimator o f tra ﬃc intens it y b y the nonparametric b o ot strap metho d for small and mo derate samples. T an, Liu, W u, Cao, and T ang [311]: Applying prob e ve hicle tra jectory , Bay esian theory , and F uzing lice nse plate recognition data and v ehicle tra jectory data for lane-based queue length estimation at signalized in tersections. V an Phu and F arhi [317]: Estimation of urban traﬃc state with prob e v ehicles. Zhang, Liu, Chen, Y u, and W ang [344]: Prop oses a cycle-based end-of-queue estima- tion metho d for queue length using sampled ve hicle tra jectory data under relativ ely low prob e v ehicle penetration rates. W ang, Huang, and Lo [324]: This pap er from the ﬁeld of traﬃc engineering Com bines sho c kw av e a nalysis and Bay esian netw orks fo r traﬃc pa r a meter estimation at signalized in tersections by conside ring queue spillbac k. 2021 Asanjarani, N azarath y , and T ay lor [20]: A broad literature surv ey of parameter and state estimation for queueing system s. 19 Bassam b o o and Ibrahim [39]: Using a com bination of queueing-theoretic analysis, real- life data ana lysis, and sim ulation, the p erformance of static and dynamic announcemen ts are analysed, and an appropriate weigh ted a ve r a ge of them is deriv ed. Basak and Choudh ury [32]: Finding a Bay es estimator of traﬃc in tensit y f o r an M/M/1 queueing mo del using data on queue size (n um b er of customers presen t in the queue) ob- serv ed a t any random p o in t in time. Comert, Amdeb erhan, Begasha w, and Cho wdh ury [79]: A Comb inatorial Approac h for Nonparametric Short-T erm Estimation of Queue Lengths using Probe V ehicles. Cruz, San tos, Oliv eira, and Quinino [91 ]: Estimation of p erformance measures in a general bulk-arriv al Mark o vian m ulti-serv er ﬁnite queue. Dieleman [101 ]: The metho d o f MLE is used in combination with Sto c hastic Approxima- tion (SA) to calibrate the arriv al parameter of a G/G/1 queue via w aiting time data. Eb ert, Dutta, Mengersen, Mira, Ruggeri, and W u [112]: Presen ts a lik eliho o d-free parameter estimation for dynamic queueing netw or ks with a case study of passenger ﬂow in an in ternational airp ort terminal. Krishnasam y , Sen, Johari, and Shakk ot tai [207]: Considers regret analysis of a serv er allo cation problem where service rat es of serv ers are unkno wn. This analysis is in the con text of m ulti-armed bandits. The work extends previous w ork: [206]. Lin, He, and Pang [218]: Queuing netw ork top ology inferenc e using passiv e end-to-end measuremen ts originated b y a single source. Mandjes and Ravn er [226]: In this pap er the authors dev ise h yp o thesis testing pro ce- dures for Lévy-driv en storage systems b y sampling of the storage lev el. Singh, A chary a, Cruz, and Quinino [294]: Prop osed a metho dology to determine the sample size for a n queueing system under the Ba y esian setup b y o bserving the num b er of customer arriv als during the service time of a customer. W alton and Xu [322]: Prov ides an extensiv e review of reinforcemen t learning ideas with a view of queueing net w ork con trol. The paper also connects ideas of adv ersarial learning to queuing net w ork concep ts fr a med in the conte xt of information uncertain t y . W ang and H onnappa [323]: Studies inference for a Cox / G / ∞ queue, sampled a t discrete time p oints . Uses approxim ate inference for maximizing a lo w er b ound for the asso ciated ﬁnite dimensional distribution. Zhao, Shen, and Liu [349]: A hidden Mark ov mo del for the estimation of correlated queues in prob e v ehicle env ironmen ts. Zhao, W ong, Zheng, and Liu [350]: Maximu m lik eliho o d estimation of prob e v ehicle p enetration rates and queue length distributions from probe vehi cle data. 2022 An tunes, J acinto, and Pac heco [11]: A discussion on statistic al inference in queueing net w o r ks with probing information. Comert and Bagasha w [80]: Cy cle-to-cycle queue length estimation from connected v ehicles with ﬁltering on primary parameters. This study impro ve accuracy of by enhancing the lo w lev el parameter estimators using ﬁltering algorithms. Li, Zheng, Ok am ura, and Dohi [213]: Hierarc hical Ba yes ian P arameter Estimation of Queueing Systems using Utilization Data. Li, Zheng, Ok am ura, and Dohi [214]: P ara meter estimation of a MAP/M/1/K queueing system using utilization data. In particular, the para meters are estimated by using the 20 maxim um lik eliho o d estimation (MLE) metho d. Ra vner [273]: Considers an M /G/ 1 queue fo r whic h the workload pro cess is observ ed p erio dically . The goal is to estimate the arriv al rate λ and the par a meters of the job-size distribution G . Singh, Ac hary a, Cruz, and Qui nino [295]: Ba y esian inference and prediction in a queueing system . Zhong, Birge, and W ard [353]: Considers a sc heduling p olicy in time-v arying m ulticlass man y serv er queues with a bandonmen t and prop oses a Learn-Then-Sc hedule alg o rithm com- p osed of a learning phase and an exploitation phase. 2023 Chen, Liu, and Hong [69]: Prop oses a nd studies an on- line learning algorithm f o r optimal con trol and pricing of a GI/GI/1 queue. Bura and Sharma [56 ]: Maxim um like liho o d and Ba y esian estimation for an M/M/1 queueing mo del with balking. Carmeli, Y om-T o v and Bo xma [59]: Analyzes fork-jo in net works and incorp orates data driv en estimation f o r suc h mo dels. Inoue, Ra vner, and Mandjes [167]: Deals with parameter estimation of a queueing system with impatien t customers and balking. A no v el algo rithm for the estimation of customer impatience is prop osed and analyzed. Luo, Deng, Chen, et. a l. [223]: This pap er f rom the ﬁeld of traﬃc engineering fo cuses on que ue length estimation based on prob e v ehicle data at signalized in tersections using a sho c kw av e approac h in the mo del. Ra vner and W ang [275]: P arameter estimation of a queueing system where customers b eha ve strategically , are able to c ho o se their arriv al times, and are at a Nash equilibrium. 21 2 Literature Analysis (up to 2017) This section w as last up dated in 201 7 and captures a classiﬁcation of the pap ers up to 2 0 17 via sev eral categories. It also serv es as a background companion to our surv ey pap er, [2 0]. Up to 201 7, the ma jority of the references we re in the format of journal and/or conference articles. A few are surv eys, textb o o ks, b o o k c hapters, Ph.D. thes es, and signiﬁcan t related materials whic h a re listed in the table b elow. Surv eys [46] [45] and the mor e recen t [20] T extb o oks [52] [240] [250] [265] [4 3 ] [340] Bo ok c hapters [44] –Chapter 2, [42 ] –Chapter 10, [140] –Section 6.7, [123] –Chapter 6, [3 2 1] –Chapter 10 Ph.D. theses [138] [271] [164] [184] [161] [3 4 8] Signiﬁcan t related materials [47] (A b o ok) 2.1 Classiﬁcation b y Mo del Mo del References M/M/1 [77] [139] [93] [17 9] [4] [286] [236] [237] [208] [1 7][16] [234] [281] [300] [71] [307] [100] [66 ] [338] [232] [303] [276] [27 7] [67] [267] [352] M/M/2 (Heterogenous Servers) [98] [113] M/M/1/K [236] [237] [6] 22 Mo del References M/M/1/ ∞ (FIFO) [14] M/M/c [224] [148] [238] [325] [283] [2 3 8] [2] [33 9] [67] M/M/c/N [192] [243] M/M/ ∞ [18] [229] k-P ar/M/1 (k-P ar denotes a mixture of k Pareto distributions) [270] M/D/1 [253] [302] M/E k /1 [176] [347] M/G/ ∞ (Random translation mo dels in general) [40] [55] [53] [269] [261] [49] [25 7] [229] [141] [2 30] [249] [330] [191] M/G/1 [93] [313] [157] [158] [170] [14 9 ] [126] [2 3 1] [99] [102] [1 68] [50 ] [172] [24] [119] [62] [27 9 ] [74] [18 8 ] [189] [193] [24 7] [346] [65] [242] [10] [67] [13 3] M/GI/1 [1] E k /G/1 [119] M/G/c [95] 23 Mo del References M/G/c/C [163] G/G/1 [316] [280] [75] [67] GI/M/1 (state-dependent arriv al rate) [92] [256] [174] GI/M/1/n [1] GI/M/c [337] [23] [165] GI/G/1 [93] [38] [263] [34 ] [3] [173] [85] [3 5 ] GI/G/2 [178] GI/G/c [202] G/G/ ∞ [269] GI/K/1 (service time has Matrix-Exp onential distribution) [21] [22] [291] [29 7] GI/GI/s [104] GI/GI/ ∞ [104] M/D/ ∞ (Deterministic service) [137] M/D/1 [224] G/D/1 [153] [152] 24 Mo del References MAP/D/1 [61] Erlang Loss System [282] Jac kson Net works or More General Netw orks [315] [27] Systems Where Little’s La w Holds [131] 2.2 Classiﬁcation b y Sampl i ng Re g ime Sampling R egime References F ull Observ ation [266] Observ ation at Discrete P oints [99] [238] [283] Probing [293] [6] [12] [27] [68] [81 ] [153] [156] [155] [185] [201] [247] [253] [10] Queue Inference E ngine [209] [1 2 7] [63] [102] [2 2 5] [12 2] [258] [96] [97] [154] 25 2.3 Classiﬁcation b y Statistical Paradigm Statistical P aradigm References Ba yes ian [245] [278] [328] [236] [237] [6 3 ] [314] [2 96] [16] [17] [15] [18] [168] [281] [82] [18 0] [19] [3 0 0] [24] [84] [62] [23] [243 ] [71] [25] [187] [272] [3 0 9] [242] [276] [277] [267] Maxim um E n tropy [151] Emphasis on the w a y of selecting sampling t ime [38] Non-parametric [82] [49] [180] [83 ] [84] [186] [257] [235] [133] Change p oin t detect ion [171] A daptive Con t rol [157] [330] Sequen tial Inference [175] [33] [319] P ert urbation analysis [160] Large Deviations [130] 26 2.4 Classiﬁcation b y Applicatio n Statistical Parad igm References T elephone Call Cen tres [54] [135] [165] [125] [19 8] [330] [26] Man ufacturing [177] [62] [165] [66] [33 0] Health Care [198] [201] [242] [333] [339] T ranspor t ation [62] [216] [219] [305] [3 4 1] [13] Economics [266] A TM [108] [102] [83] [85] [62 ] Comm unication/T elecomm unication Net works [127] [27] [29] [23 3] [241] Net work T raﬃc Mo delling [292] [227] [61] [220] [3 0] [301] [309] 27 References [1] V.M. 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