Scheduling for today's computer systems: bridging theory and practice

Scheduling is a fundamental technique for improving performance in computer systems. From web servers to routers to operating systems, how the bottleneck device is scheduled has an enormous impact on the performance of the system as a whole. Given the immense literature studying scheduling, it is easy to think that we already understand enough about scheduling. But, modern computer system designs have highlighted a number of disconnects between traditional analytic results and the needs of system designers. In particular, the idealized policies, metrics, and models used by analytic researchers do not match the policies, metrics, and scenarios that appear in real systems. The goal of this thesis is to take a step towards modernizing the theory of scheduling in order to provide results that apply to today's computer systems, and thus ease the burden on system designers. To accomplish this goal, we provide new results that help to bridge each of the disconnects mentioned above. We will move beyond the study of idealized policies by introducing a new analytic framework where the focus is on scheduling heuristics and techniques rather than individual policies. By moving beyond the study of individual policies, our results apply to the complex hybrid policies that are often used in practice. For example, our results enable designers to understand how the policies that favor small job sizes are affected by the fact that real systems only have estimates of job sizes. In addition, we move beyond the study of mean response time and provide results characterizing the distribution of response time and the fairness of scheduling policies. These results allow us to understand how scheduling affects QoS guarantees and whether favoring small job sizes results in large job sizes being treated unfairly. Finally, we move beyond the simplified models traditionally used in scheduling research and provide results characterizing the effectiveness of scheduling in multiserver systems and when users are interactive. These results allow us to answer questions about the how to design multiserver systems and how to choose a workload generator when evaluating new scheduling designs.

[1]  Sidney L. Hantler,et al.  On a Preemptive Markovian Queue with Multiple Servers and Two Priority Classes , 1992, Math. Oper. Res..

[2]  LiuZhen,et al.  Traffic model and performance evaluation of Web servers , 2001 .

[3]  Ward Whitt,et al.  Predicting Response Times in Processor-Sharing Queues , 2000 .

[4]  Adam Wierman,et al.  Fairness and classifications , 2007, PERV.

[5]  Scott Shenker,et al.  Analysis and simulation of a fair queueing algorithm , 1989, SIGCOMM '89.

[6]  Paul Barford,et al.  Generating representative Web workloads for network and server performance evaluation , 1998, SIGMETRICS '98/PERFORMANCE '98.

[7]  Adam Wierman,et al.  On the effect of inexact size information in size based policies , 2006, PERV.

[8]  Onno Boxma,et al.  SOJOURN TIME TAILS IN THE M/D/1 PROCESSOR SHARING QUEUE , 2006, Probability in the Engineering and Informational Sciences.

[9]  Adam Wierman,et al.  Open Versus Closed: A Cautionary Tale , 2006, NSDI.

[10]  Dimitris Bertsimas,et al.  The Distributional Little's Law and Its Applications , 1995, Oper. Res..

[11]  Zhen Liu,et al.  Traffic model and performance evaluation of Web servers , 2001, Perform. Evaluation.

[12]  Adam Wierman,et al.  A recursive analysis technique for multi-dimensionally infinite Markov chains , 2004, PERV.

[13]  J. F. Meyer,et al.  Probabilistic modeling , 1995 .

[14]  B. Avi-Itzhak,et al.  On measuring fairness in queues , 2004, Advances in Applied Probability.

[15]  Andrei Sleptchenko,et al.  Multi-class, multi-server queues with non-preemptive priorities , 2003 .

[16]  Vishal Misra,et al.  PBS: a unified priority-based scheduler , 2007, SIGMETRICS '07.

[17]  Mor Harchol-Balter.  Task assignment with unknown duration , 2002, JACM.

[18]  Richard H. Davis.  Waiting-Time Distribution of a Multi-Server, Priority Queuing System , 1966, Oper. Res..

[19]  R. Schassberger,et al.  A new approach to the M/G/1 processor-sharing queue , 1984, Advances in Applied Probability.

[20]  Eric J. Friedman,et al.  Fairness and efficiency in web server protocols , 2003, SIGMETRICS '03.

[21]  Walter Willinger,et al.  On the Self-Similar Nature of Ethernet Traffic ( extended version ) , 1995 .

[22]  R. Núñez Queija,et al.  TCP as an Implementation of Age-Based Scheduling: Fairness and Performance , 2006, Proceedings IEEE INFOCOM 2006. 25TH IEEE International Conference on Computer Communications.

[23]  Peter G. Taylor,et al.  Calculating the equilibrium distribution in level dependent quasi-birth-and-death processes , 1995 .

[24]  M. Hui,et al.  How Does Waiting Duration Information Influence Customers' Reactions to Waiting for Services?1 , 1996 .

[25]  W. Sandmann,et al.  A discrimination frequency based queueing fairness measure with regard to job seniority and service requirement , 2005, Next Generation Internet Networks, 2005.

[26]  Thomas Bonald,et al.  Statistical bandwidth sharing: a study of congestion at flow level , 2001, SIGCOMM.

[27]  Onno Boxma,et al.  Some time-dependent properties of symmetric M/G/1 queues , 2005 .

[28]  P. Ney,et al.  Functions of probability measures , 1973 .

[29]  Ward Whitt,et al.  A review ofL=λW and extensions , 1991, Queueing Syst. Theory Appl..

[30]  Sem C. Borst,et al.  The equivalence between processor sharing and service in random order , 2003, Oper. Res. Lett..

[31]  Peter Druschel,et al.  Measuring the capacity of a Web server under realistic loads , 1999, World Wide Web.

[32]  Adam Wierman,et al.  Formalizing SMART scheduling , 2004, PERV.

[33]  B. A. Taylor,et al.  Analysis of a non-preemptive priority multiserver queue , 1988, Advances in Applied Probability.

[34]  Adam Wierman,et al.  Understanding the slowdown of large jobs in an M/GI/1 system , 2002, PERV.

[35]  Abhay Parekh,et al.  A generalized processor sharing approach to flow control in integrated services networks-the single node case , 1992, [Proceedings] IEEE INFOCOM '92: The Conference on Computer Communications.

[36]  Daniel A. Menascé,et al.  Scaling for E-Business: Technologies, Models, Performance, and Capacity Planning , 2000 .

[37]  S. F. Yashkov.  Mathematical problems in the theory of shared-processor systems , 1992 .

[38]  Adam Wierman,et al.  How to Determine a Good Multi-Programming Level for External Scheduling , 2006, 22nd International Conference on Data Engineering (ICDE'06).

[39]  Benjamin Avi-Itzhak,et al.  Fairness considerations of scheduling in multi-server and multi-queue systems , 2006, valuetools '06.

[40]  J. B. M. Van Doremalen.  A note on "analysis of a finite capacity nonpreemptive priority queue" by Kapadia, Kazmi and Mitchell , 1986, Comput. Oper. Res..

[41]  Urtzi Ayesta,et al.  Two-level processor-sharing scheduling disciplines: mean delay analysis , 2004, SIGMETRICS '04/Performance '04.

[42]  D. Korshunov.  On distribution tail of the maximum of a random walk , 1997 .

[43]  B. Avi-Itzhak,et al.  Fairness Considerations in Multi-Server and Multi-Queue Systems , 2005 .

[44]  Sheldon M. Ross,et al.  Introduction to probability models , 1975 .

[45]  Frank Kelly,et al.  Rate control for communication networks: shadow prices, proportional fairness and stability , 1998, J. Oper. Res. Soc..

[46]  William L. Maxwell,et al.  Theory of scheduling , 1967 .

[47]  Tareghian,et al.  Designing a production line through optimisation of M/G/c* using simulation , 2001 .

[48]  Ajay D. Kshemkalyani,et al.  SWIFT: scheduling in web servers for fast response time , 2003, Second IEEE International Symposium on Network Computing and Applications, 2003. NCA 2003..

[49]  Adam Wierman,et al.  Modeling TCP-vegas under on/off traffic , 2003, PERV.

[50]  Mor Harchol-Balter,et al.  Improving preemptive prioritization via statistical characterization of OLTP locking , 2005, 21st International Conference on Data Engineering (ICDE'05).

[51]  Mark S. Squillante,et al.  Analysis of task assignment with cycle stealing under central queue , 2003, 23rd International Conference on Distributed Computing Systems, 2003. Proceedings..

[52]  D. F. Hays,et al.  Table of Integrals, Series, and Products , 1966 .

[53]  J. Teugels,et al.  On the asymptotic behaviour of the distributions of the busy period and service time in M/G/1 , 1980, Journal of Applied Probability.

[54]  Adam Wierman,et al.  Multi-Server Queueing Systems with Multiple Priority Classes , 2005, Queueing Syst. Theory Appl..

[55]  L. Flatto.  The waiting time distribution for the random order service $M/M/1$ queue , 1997 .

[56]  T. Ott.  THE SOJOURN-TIME DISTRIBUTION IN THE M/G/1 QUEUE , 1984 .

[57]  D. B. Cline,et al.  Intermediate Regular and Π Variation , 1994 .

[58]  Arzad Alam Kherani,et al.  Sojourn times in (discrete) time shared systems and their continuous time limits , 2006, valuetools '06.

[59]  Predrag R. Jelenkovic,et al.  Resource sharing with subexponential distributions , 2002, Proceedings.Twenty-First Annual Joint Conference of the IEEE Computer and Communications Societies.

[60]  Edward P. C. Kao,et al.  Modelling a multiprocessor system with preemptive priorities , 1991 .

[61]  Michel Mandjes,et al.  Large deviations of sojourn times in processor sharing queues , 2006, Queueing Syst. Theory Appl..

[62]  Adam Wierman,et al.  An improved upper bound for the pebbling threshold of the n-path , 2004, Discret. Math..

[63]  Junshan Zhang,et al.  Size-aided opportunistic scheduling in wireless networks , 2003, GLOBECOM '03. IEEE Global Telecommunications Conference (IEEE Cat. No.03CH37489).

[64]  Sheldon M. Ross,et al.  Stochastic Processes , 2018, Gauge Integral Structures for Stochastic Calculus and Quantum Electrodynamics.

[65]  Werner Almesberger,et al.  Linux Network Traffic Control -- Implementation Overview , 1999 .

[66]  Adam Wierman,et al.  The Foreground-Background queue: A survey , 2008, Perform. Evaluation.

[67]  Abraham Silberschatz,et al.  Operating System Concepts, 5th Edition , 1994 .

[68]  Luigi Rizzo,et al.  Dummynet: a simple approach to the evaluation of network protocols , 1997, CCRV.

[69]  Mor Harchol-Balter,et al.  On Choosing a Task Assignment Policy for a Distributed Server System , 1998, J. Parallel Distributed Comput..

[70]  Nick G. Duffield,et al.  Large deviations, the shape of the loss curve, and economies of scale in large multiplexers , 1995, Queueing Syst. Theory Appl..

[71]  Gustavo de Veciana,et al.  Bandwidth sharing: the role of user impatience , 2001, GLOBECOM'01. IEEE Global Telecommunications Conference (Cat. No.01CH37270).

[72]  Allen B. Downey,et al.  A parallel workload model and its implications for processor allocation , 1996, Proceedings. The Sixth IEEE International Symposium on High Performance Distributed Computing (Cat. No.97TB100183).

[74]  Willy Zwaenepoel,et al.  Flash: An efficient and portable Web server , 1999, USENIX Annual Technical Conference, General Track.

[75]  David L. Peterson.  Data Center I/O Patterns And Power Laws , 1996, Int. CMG Conference.

[76]  Patrick Brown,et al.  Comparing FB and PS scheduling policies , 2006, PERV.

[77]  Pablo Molinero-Fernández,et al.  Systems with multiple servers under heavy-tailed workloads , 2005, Perform. Evaluation.

[78]  Mor Harchol-Balter,et al.  Exploiting process lifetime distributions for dynamic load balancing , 1995, SIGMETRICS.

[79]  Leonard Kleinrock,et al.  Queueing Systems: Volume I-Theory , 1975 .

[80]  Rajmohan Rajaraman,et al.  Online scheduling to minimize average stretch , 1999, 40th Annual Symposium on Foundations of Computer Science (Cat. No.99CB37039).

[81]  Nikhil Bansal,et al.  Handling load with less stress , 2006, Queueing Syst. Theory Appl..

[82]  Abhay Parekh,et al.  A generalized processor sharing approach to flow control in integrated services networks: the single-node case , 1993, TNET.

[83]  Anshu Aggarwal,et al.  The Inktomi climate lab: an integrated environment for analyzing and simulating customer network traffic , 2001, IMW '01.

[84]  Adam Wierman,et al.  Nearly insensitive bounds on SMART scheduling , 2005, SIGMETRICS '05.

[85]  Gennady Samorodnitsky,et al.  Subexponentiality of the product of independent random variables , 1994 .

[86]  Takeshi Nishida.  Approximate Analysis for Heterogeneous Multiprocessor Systems with Priority Jobs , 1992, Perform. Evaluation.

[87]  Amin Vahdat,et al.  MediSyn: a synthetic streaming media service workload generator , 2003, NOSSDAV '03.

[88]  Dipankar Raychaudhuri,et al.  Size matters: size-based scheduling for MPEG-4 over wireless channels , 2003, IS&T/SPIE Electronic Imaging.

[89]  Upendra Dave,et al.  Applied Probability and Queues , 1987 .

[90]  John J. Horton,et al.  A Theory of Social Justice? , 1991, Utilitas.

[91]  M.F.M. Nuijens.  The Foreground-Background Queue , 2004 .

[92]  Vivek Tiwari,et al.  Reducing power in high-performance microprocessors , 1998, Proceedings 1998 Design and Automation Conference. 35th DAC. (Cat. No.98CH36175).

[93]  Mor Harchol-Balter,et al.  Analysis of SRPT scheduling: investigating unfairness , 2001, SIGMETRICS '01.

[94]  Herlinde Leemans,et al.  Queue lengths and waiting times in the two-class two-server queue with nonpreemptive heterogeneous priority structures , 1998 .

[95]  J. Cohen.  SOME RESULTS ON REGULAR VARIATION FOR DISTRIBUTIONS IN QUEUEING AND FLUCTUATION THEORY , 1973 .

[96]  Masayuki Murata,et al.  Comparisons of packet scheduling algorithms for fair service among connections on the Internet , 2000, Proceedings IEEE INFOCOM 2000. Conference on Computer Communications. Nineteenth Annual Joint Conference of the IEEE Computer and Communications Societies (Cat. No.00CH37064).

[97]  Werner Sandmann,et al.  Analysis of a queueing fairness measure , 2006, MMB.

[98]  Adam Wierman,et al.  Classifying scheduling policies with respect to higher moments of conditional response time , 2005, SIGMETRICS '05.

[99]  Mor Harchol-Balter,et al.  Analysis of multi-server systems via dimensionality reduction of markov chains , 2005 .

[100]  U. Yechiali,et al.  Waiting Times in the Non-Preemptive Priority M/M/c Queue , 1985 .

[101]  Martin F. Arlitt,et al.  Web server workload characterization: the search for invariants , 1996, SIGMETRICS '96.

[102]  M. Shalmon.  Analysis of the GI/GI/1 Queue and its Variations via the LCFS Preemptive Resume Discipline and Its Random Walk Interpretation , 1988, Probability in the Engineering and Informational Sciences.

[103]  Adam Wierman,et al.  A unified framework for modeling TCP-Vegas, TCP-SACK, and TCP-Reno , 2003, 11th IEEE/ACM International Symposium on Modeling, Analysis and Simulation of Computer Telecommunications Systems, 2003. MASCOTS 2003..

[104]  Randolph Nelson,et al.  Probability, Stochastic Processes, and Queueing Theory , 1995 .

[105]  Guillaume Urvoy-Keller,et al.  Analysis of LAS scheduling for job size distributions with high variance , 2003, SIGMETRICS '03.

[106]  C. Amza,et al.  Specification and implementation of dynamic Web site benchmarks , 2002, 2002 IEEE International Workshop on Workload Characterization.

[107]  Adam Wierman,et al.  Preventing Large Sojourn Times Using SMART Scheduling , 2008, Oper. Res..

[108]  Rudesindo Núñez-Queija,et al.  Queues with Equally Heavy Sojourn Time and Service Requirement Distributions , 2002 .

[109]  Gustavo de Veciana,et al.  Enhancing both network and user performance for networks supporting best effort traffic , 2004, IEEE/ACM Transactions on Networking.

[110]  Anthony Unwin,et al.  Reversibility and Stochastic Networks , 1980 .

[111]  M. Kendall,et al.  The advanced theory of statistics , 1945 .

[112]  Azer Bestavros,et al.  Self-similarity in World Wide Web traffic: evidence and possible causes , 1996, SIGMETRICS '96.

[113]  Peter A. Dinda,et al.  Size-based scheduling policies with inaccurate scheduling information , 2004, The IEEE Computer Society's 12th Annual International Symposium on Modeling, Analysis, and Simulation of Computer and Telecommunications Systems, 2004. (MASCOTS 2004). Proceedings..

[114]  Chang Woo Yang,et al.  Delay asymptotics of the SRPT scheduler , 2004, 2004 43rd IEEE Conference on Decision and Control (CDC) (IEEE Cat. No.04CH37601).

[115]  Jordi Torres,et al.  Session-based adaptive overload control for secure dynamic Web applications , 2005, 2005 International Conference on Parallel Processing (ICPP'05).

[116]  Adam Wierman,et al.  Scheduling in polling systems , 2007, Perform. Evaluation.

[117]  Trevor N. Mudge,et al.  Power: A First-Class Architectural Design Constraint , 2001, Computer.

[118]  Matthieu van der Heijden,et al.  An Exact Solution for the State Probabilities of the Multi-Class, Multi-Server Queue with Preemptive Priorities , 2005, Queueing Syst. Theory Appl..

[119]  Michael A. Bender,et al.  Flow and stretch metrics for scheduling continuous job streams , 1998, SODA '98.

[120]  Linus Schrage,et al.  The Queue M/G/1 with the Shortest Remaining Processing Time Discipline , 1966, Oper. Res..

[121]  Ravishankar K. Iyer,et al.  Geist: a generator for e-commerce & internet server traffic , 2001, 2001 IEEE International Symposium on Performance Analysis of Systems and Software. ISPASS..

[122]  I. S. Gradshteyn,et al.  Table of Integrals, Series, and Products , 1976 .

[123]  Alexandre Proutière,et al.  Statistical bandwidth sharing: a study of congestion at flow level , 2001, SIGCOMM.

[124]  Paul Farmer,et al.  Medicine and social justice. , 1995, America.

[125]  James E. Stafford,et al.  Symbolic Computation of Moments in Priority Queues , 2002, INFORMS J. Comput..

[126]  Fabrice Guillemin,et al.  Tail asymptotics for processor-sharing queues , 2004, Advances in Applied Probability.

[127]  Peter E. Sholander,et al.  IPB: an Internet Protocol Benchmark using simulated traffic , 1998, Proceedings. Sixth International Symposium on Modeling, Analysis and Simulation of Computer and Telecommunication Systems (Cat. No.98TB100247).

[128]  Onno J. Boxma,et al.  The busy period in the fluid queue , 1998, SIGMETRICS '98/PERFORMANCE '98.

[129]  Adam Wierman,et al.  A note on comparing response times in the M/GI/1/FB and M/GI/1/PS queues , 2004, Oper. Res. Lett..

[130]  Sem C. Borst,et al.  The impact of the service discipline on delay asymptotics , 2003, Perform. Evaluation.

[131]  W. Whitt,et al.  Heavy-traffic asymptotic expansions for the asymptotic decay rates in the BMAP/G/1 queue , 1994 .

[132]  Alexandre Proutière,et al.  Insensitive Bandwidth Sharing in Data Networks , 2003, Queueing Syst. Theory Appl..

[133]  David Mosberger,et al.  httperf—a tool for measuring web server performance , 1998, PERV.

[134]  M. A. Aczel.  Letter to the Editor—The Effect of Introducing Priorities , 1960 .

[135]  Avishai Mandelbaum,et al.  On Pooling in Queueing Networks , 1998 .

[136]  Adam Wierman,et al.  Asymptotic convergence of scheduling policies with respect to slowdown , 2002, Perform. Evaluation.

[137]  Xiuli Chao,et al.  Several Results on the Design of Queueing Systems , 2000, Oper. Res..

[138]  N. Bingham,et al.  Asymptotic properties of supercritical branching processes I: The Galton-Watson process , 1974, Advances in Applied Probability.

[139]  Mikko Alava,et al.  Branching Processes , 2009, Encyclopedia of Complexity and Systems Science.

[140]  Ray Jain,et al.  The art of computer systems performance analysis - techniques for experimental design, measurement, simulation, and modeling , 1991, Wiley professional computing.

[141]  Timothy I. Matis,et al.  Using Cumulant Functions in Queueing Theory , 2002, Queueing Syst. Theory Appl..

[142]  Joel M. Calabrese.  Optimal workload allocation in open networks of multiserver queues , 1992 .

[143]  Walter Willinger,et al.  Self-Similar Network Traffic and Performance Evaluation , 2000 .

[144]  Benjamin Avi-Itzhak,et al.  Server sharing with a limited number of service positions and symmetric queues , 1987 .

[145]  Carey L. Williamson,et al.  Quantifying the properties of SRPT scheduling , 2003, 11th IEEE/ACM International Symposium on Modeling, Analysis and Simulation of Computer Telecommunications Systems, 2003. MASCOTS 2003..

[146]  Guillaume Urvoy-Keller,et al.  Performance Models for LAS-based Scheduling Disciplines in a Packet Switched Network , 2004 .

[147]  Dan Rubenstein,et al.  PBS: A Unified Priority-Based CPU Scheduler , 2007 .

[148]  J. Shanthikumar,et al.  On extremal service disciplines in single-stage queueing systems , 1990, Journal of Applied Probability.

[149]  Duane Wessels,et al.  High‐performance benchmarking with Web Polygraph , 2004, Softw. Pract. Exp..

[150]  Laura A. Dabbish,et al.  Labeling images with a computer game , 2004, AAAI Spring Symposium: Knowledge Collection from Volunteer Contributors.

[151]  Alan Scheller-Wolf.  Necessary and Sufficient Conditions for Delay Moments in FIFO Multiserver Queues with an Application Comparing s Slow Servers with One Fast One , 2003, Oper. Res..

[152]  Edward P. C. Kao,et al.  Analysis of nonpreemptive priority queues with multiple servers and two priority classes , 1999, Eur. J. Oper. Res..

[153]  Anja Feldmann,et al.  Dynamics of IP traffic: a study of the role of variability and the impact of control , 1999, SIGCOMM '99.

[154]  Lawrence W. Dowdy,et al.  On the Applicability of Using Multiprogramming Level Distributions , 1985, SIGMETRICS.

[155]  Jeffrey P. Buzen,et al.  The response times of priority classes under preemptive resume in M/G/m queues , 1984, SIGMETRICS '84.

[156]  A. Stolyar,et al.  LARGEST WEIGHTED DELAY FIRST SCHEDULING: LARGE DEVIATIONS AND OPTIMALITY , 2001 .

[157]  M. F. Kazmi,et al.  Analysis of a finite capacity non preemptive priority queue , 1984, Comput. Oper. Res..

[158]  Adam Wierman,et al.  How many servers are best in a dual-priority M/PH/k system? , 2006, Perform. Evaluation.

[159]  Alan Cobham,et al.  Priority Assignment in Waiting Line Problems , 1954, Oper. Res..

[160]  Allen B. Downey,et al.  Evidence for long-tailed distributions in the internet , 2001, IMW '01.

[161]  Adam Wierman,et al.  Competitive Analysis of M/GI/1 Queueing Policies , 2003 .

[162]  M. Malik,et al.  Operating Systems , 1992, Lecture Notes in Computer Science.

[163]  Ludmila Cherkasova.  Scheduling Strategy to improve Response Time for Web Applications , 1998, HPCN Europe.

[164]  J. W. Roberts,et al.  A survey on statistical bandwidth sharing , 2004, Comput. Networks.

[165]  Catherine Rosenberg,et al.  Tail Asymptotics for HOL Priority Queues Handling a Large Number of Independent Stationary Sources , 2002, Queueing Syst. Theory Appl..

[166]  R. Fisher.  The Advanced Theory of Statistics , 1943, Nature.

[167]  Urtzi Ayesta,et al.  M/G/1/MLPS compared to M/G/1/PS , 2005, Oper. Res. Lett..

[168]  Alan Scheller-Wolf,et al.  Analysis of cycle stealing with switching cost , 2003, SIGMETRICS '03.

[169]  V. Chistyakov.  A Theorem on Sums of Independent Positive Random Variables and Its Applications to Branching Random Processes , 1964 .

[170]  Mor Harchol-Balter,et al.  Priority mechanisms for OLTP and transactional Web applications , 2004, Proceedings. 20th International Conference on Data Engineering.

[171]  Edward P. C. Kao,et al.  Computing Steady-State Probabilities of a Nonpreemptive Priority Multiserver Queue , 1990, INFORMS J. Comput..

[172]  B. Kahn,et al.  How Tolerable is Delay? Consumers’ Evaluations of Internet Web Sites after Waiting , 1998 .

[173]  Carey L. Williamson,et al.  Simulation evaluation of hybrid SRPT scheduling policies , 2004, The IEEE Computer Society's 12th Annual International Symposium on Modeling, Analysis, and Simulation of Computer and Telecommunications Systems, 2004. (MASCOTS 2004). Proceedings..

[174]  Ward Whitt,et al.  Numerical Inversion of Laplace Transforms of Probability Distributions , 1995, INFORMS J. Comput..

[175]  Bert Zwart,et al.  A large-deviations analysis of the GI/GI/1 SRPT queue , 2005, Queueing Syst. Theory Appl..

[176]  Mor Harchol-Balter,et al.  A Closed-Form Solution for Mapping General Distributions to Minimal PH Distributions , 2003, Computer Performance Evaluation / TOOLS.

[177]  M. Mandjes,et al.  SOJOURN TIMES IN THE M/G/1 FB QUEUE WITH LIGHT-TAILED SERVICE TIMES , 2003, Probability in the Engineering and Informational Sciences.

[178]  D. Kendall.  Stochastic Processes Occurring in the Theory of Queues and their Analysis by the Method of the Imbedded Markov Chain , 1953 .

[179]  Mor Harchol-Balter,et al.  Web servers under overload: How scheduling can help , 2006, TOIT.

[180]  A. Pakes.  ON THE TAILS OF WAITING-TIME DISTRIBUTIONS , 1975 .

[181]  Alexandre Proutière,et al.  On performance bounds for the integration of elastic and adaptive streaming flows , 2004, SIGMETRICS '04/Performance '04.

[182]  Adam Wierman,et al.  Tail asymptotics for policies favoring short jobs in a many-flows regime , 2006, SIGMETRICS '06/Performance '06.

[183]  S. F. Yashkov,et al.  Processor-sharing queues: Some progress in analysis , 1987, Queueing Syst. Theory Appl..

[184]  P. Schatte.  On conditional busy periods in queues , 1983 .

[185]  Thomas E. Phipps.  Machine Repair as a Priority Waiting-Line Problem , 1956 .

[186]  M. Meerschaert.  Regular Variation in R k , 1988 .

[187]  O. Hawaleshka,et al.  On The Interrelationship Between Semi-Open and Closed Queueing Network Models For Flexible Manufacturing System , 1987 .

[188]  J. George Shanthikumar,et al.  Scheduling Multiclass Single Server Queueing Systems to Stochastically Maximize the Number of Successful Departures , 1989, Probability in the Engineering and Informational Sciences.

[189]  Ward Whitt,et al.  A Unified Framework for Numerically Inverting Laplace Transforms , 2006, INFORMS J. Comput..

[190]  Martin Arlitt,et al.  A workload characterization study of the 1998 World Cup Web site , 2000, IEEE Netw..

[191]  Sem C. Borst,et al.  Sojourn time asymptotics in processor-sharing queues , 2006, Queueing Syst. Theory Appl..

[192]  Srinivasan Seshan,et al.  The effects of wide-area conditions on WWW server performance , 2001, SIGMETRICS '01.

[193]  Linus Schrage,et al.  Letter to the Editor - A Proof of the Optimality of the Shortest Remaining Processing Time Discipline , 1968, Oper. Res..

[194]  Benjamin Avi-Itzhak,et al.  A resource allocation queueing fairness measure: properties and bounds , 2007, Queueing Syst. Theory Appl..

[195]  Peter A. Dinda,et al.  Looking at the server side of peer-to-peer systems , 2004, LCR.

[196]  P. Schatte.  The M/GI/1 queue as limit of closed queueing systems , 1984 .

[197]  Alexandre Proutière,et al.  Insensitivity in processor-sharing networks , 2002, Perform. Evaluation.

[198]  Peter J. B. King,et al.  Multiprocessor systems with preemptive priorities , 1981, Perform. Evaluation.

[199]  Gennady Samorodnitsky,et al.  Activity periods of an infinite server queue and performance of certain heavy tailed fluid queues , 1999, Queueing Syst. Theory Appl..

[200]  Tom Burr,et al.  Introduction to Matrix Analytic Methods in Stochastic Modeling , 2001, Technometrics.

[201]  Sally Floyd,et al.  Wide-area traffic: the failure of Poisson modeling , 1994 .

[202]  Nikhil Bansal.  On the average sojourn time under M/M/1/SRPT , 2003, PERV.

[203]  Mor Harchol-Balter,et al.  Size-based scheduling to improve web performance , 2003, TOCS.

[204]  Adam Wierman,et al.  Classifying scheduling policies with respect to unfairness in an M/GI/1 , 2003, SIGMETRICS '03.

[205]  Shaler Stidham,et al.  On the Optimality of Single-Server Queuing Systems , 1970, Oper. Res..

[206]  Benjamin Avi-Itzhak,et al.  A resource-allocation queueing fairness measure , 2004, SIGMETRICS '04/Performance '04.

[207]  R. Srikant,et al.  Many-Sources Delay Asymptotics with Applications to Priority Queues , 2001, Queueing Syst. Theory Appl..

[208]  A. P. Zwart,et al.  Sojourn time asymptotics in the M/G/1 processor sharing queue , 1998, Queueing Syst. Theory Appl..

[209]  Venkat Anantharam,et al.  How large delays build up in a GI/G/1 queue , 1989, Queueing Syst. Theory Appl..

[210]  Laurent Massoulié,et al.  Impact of fairness on Internet performance , 2001, SIGMETRICS '01.

[211]  Jeffrey P. Buzen,et al.  The Response Times of Priority Classes under Preemptive Resume in M/M/m Queues , 1983, Oper. Res..

[212]  Peter A. Dinda,et al.  Effects and implications of file size/service time correlation on Web server scheduling policies , 2005, 13th IEEE International Symposium on Modeling, Analysis, and Simulation of Computer and Telecommunication Systems.

[213]  SchroederBianca,et al.  Size-based scheduling to improve web performance , 2003 .

[214]  B. Ngo,et al.  Analysis of a pre-emptive priority M/M/c model with two types of customers and restriction , 1990 .

[215]  Murat Yuksel,et al.  Workload Generation for ns Simulations of Wide Area Networks and the Internet , 2000 .

[216]  Philip M. Morse,et al.  Queues, Inventories, And Maintenance , 1958 .

[217]  Lajos Takács,et al.  A Single-Server Queue with Poisson Input , 1962 .

[218]  Michael Shalmon.  Explicit formulas for the variance of conditioned sojourn times in M/D/1-PS , 2007, Oper. Res. Lett..

[219]  Manish Gupta,et al.  Power-Aware Microarchitecture: Design and Modeling Challenges for Next-Generation Microprocessors , 2000, IEEE Micro.

[220]  Ward Whitt,et al.  A Nonstationary Offered-Load Model for Packet Networks , 2001, Telecommun. Syst..

[221]  Ronald W. Wolff,et al.  Stochastic Modeling and the Theory of Queues , 1989 .

[222]  Robert B. Cooper,et al.  Queueing systems, volume II: computer applications : By Leonard Kleinrock. Wiley-Interscience, New York, 1976, xx + 549 pp. , 1977 .

[223]  Shawn Ostermann,et al.  The trafgen traffic generator , 2002 .

[224]  C. Klüppelberg,et al.  Tail behaviour of the busy period of a GI/GI/1 queue with subexponential service times , 2004 .

[225]  Anees Shaikh,et al.  Load-sensitive routing of long-lived IP flows , 1999, SIGCOMM '99.

[226]  Michael Pinedo,et al.  Scheduling: Theory, Algorithms, and Systems , 1994 .