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Queueing and Scheduling Bridging the Gap Gideon Weiss University of Haifa CIRM Marseilles 12 Queueing and Scheduling Bridging the Gap Gideon Weiss University of Haifa CIRM Marseilles 12 -16 May, 2008

Gideon Weiss, University of Haifa, Scheduling and Queues, © 2008 1 Gideon Weiss, University of Haifa, Scheduling and Queues, © 2008 1

Outline of the talk: We survey of some stochastic scheduling and queueing results and Outline of the talk: We survey of some stochastic scheduling and queueing results and put them in context • Single machine - performance evaluation - batch vs stream • Parallel machines - pooling and pipelining • Job shop scheduling - the 10 x 10 problem - demo of job shop simulator • Fluid control of MCQN with IVQs joint work with Anat Kopzon and Yoni Nazarathy Gideon Weiss, University of Haifa, Scheduling and Queues, © 2008 2

Single Machine Scheduling - Batch of N jobs, Minimize flowtime, SPT optimal Three proofs Single Machine Scheduling - Batch of N jobs, Minimize flowtime, SPT optimal Three proofs #1: Pairwise interchange: #2: Calculate cost of each job: #3: In every pair of jobs one waits for the other (Matloff ‘ 86) Performance: ~30% savings Gideon Weiss, University of Haifa, Scheduling and Queues, © 2008 3

Single Machine Scheduling - Stream Arrival stream (release times) at NP-hard Work conservation: amount Single Machine Scheduling - Stream Arrival stream (release times) at NP-hard Work conservation: amount of work remaining independent of schedule Min Makespan = Max Utilization = Non-Idling = Work Conserving Min Flowtime = Min Waiting = Min Queue Length (Little’s Law) SEPT optimal: Waiting is done almost exclusively by long jobs F(List)/F(SEPT) = Longest jobs/Average jobs = Unbounded Gideon Weiss, University of Haifa, Scheduling and Queues, © 2008 4

Single Machine Scheduling - Computer Science Computers usually allocate processor sharing (round robin) Sojourn Single Machine Scheduling - Computer Science Computers usually allocate processor sharing (round robin) Sojourn is Proporional to Length - Fair (same for LCFS) In internet: mice and elephants. Modern work (Mor Harchol Balther, Wierman, Scheller Wolfe) shows vast improvement by using SRPT with no penalty for long jobs Priority Rules - Bandit Problems and Gittins Index Preemptive scheduling, stochastic jobs, Gittins index (Klimov) optimal Whittle index - for restless bandits Nino Mora - Marginal Productivity Index Protocols ALOHA, TCP (vast literature), RED, etc. Gideon Weiss, University of Haifa, Scheduling and Queues, © 2008 5

Single Machine Lessons Performance evaluation and comparison of policies Stream scheduling: same optimal policy, Single Machine Lessons Performance evaluation and comparison of policies Stream scheduling: same optimal policy, but very different costs Makespan = Utilization Flowtime = Queue length = Waiting time Gideon Weiss, University of Haifa, Scheduling and Queues, © 2008 6

Parallel Machine Scheduling - Batch Flowtime: SPT optimal. Stochastic jobs: SEPT optimal (for stochastically Parallel Machine Scheduling - Batch Flowtime: SPT optimal. Stochastic jobs: SEPT optimal (for stochastically comparable jobs) Weber Varaiya Walrand ‘ 86 Makespan: NP-hard, but in practice list is almost optimal (end effect) Weighted Flowtime: NP-hard. WF(Smith Rule)/WF(Optimum)≤ 1. 2 For stochastic jobs, worst case performance of Smith Rule (c -Rule) ≤ 1+1/N 2 W ‘ 92 Lesson: With less information optimum may get closer to heuristic solution Gideon Weiss, University of Haifa, Scheduling and Queues, © 2008 7

Parallel Machine Scheduling - Batch Stochastic result: Expected flowtime for i. i. d. jobs: Parallel Machine Scheduling - Batch Stochastic result: Expected flowtime for i. i. d. jobs: N jobs, M+1 machines: Lesson: Resource Pooling Gideon Weiss, University of Haifa, Scheduling and Queues, © 2008 8

Parallel Machine Scheduling - Stream In heavy traffic, parallel machines exhibit resource pooling. c Parallel Machine Scheduling - Stream In heavy traffic, parallel machines exhibit resource pooling. c -Rule or Gittins index scheduling are nearly optimal Lesson: Resource Pooling Gideon Weiss, University of Haifa, Scheduling and Queues, © 2008 9

Gideon Weiss, University of Haifa, Scheduling and Queues, © 2008 10 Gideon Weiss, University of Haifa, Scheduling and Queues, © 2008 10

Job Shop Scheduling - Minimum Makespan, Fluid approach: Consider the famous 10 x 10 Job Shop Scheduling - Minimum Makespan, Fluid approach: Consider the famous 10 x 10 job shop, with 930 optimum. If jobs were fluid and machines infinitely divisible: 100 activities, calculate total work for each of 10 machines, find bottleneck, T*=631 Allocate machine capacities to work at constant rate on all 100 activities Bottleneck fully utilized, others only partially. Gideon Weiss, University of Haifa, Scheduling and Queues, © 2008 11

Multi-project Scheduling Project scheduling one usually thinks of CPM, and adds resource constraints But Multi-project Scheduling Project scheduling one usually thinks of CPM, and adds resource constraints But if there are many similar projects, and dedicated teams to do various types of activities, a queueing view may be better. Example 1: Agricultural extension In the 70 s I worked for Tahal - Israel Water Planning. We were doing agricultural extension in Qazwin region of Iran, Dozens of villages, similar projects, teams of instructors, equipment, etc. Example 2: US naval shipyard - refurbishing destroyers and submarines Rob Leachman and Steve Hackman developed a scheme to schedule and Control these Example 3: Israel Electric Corporation - (Lemberg, Avi Mandelbaum, W ‘ 93) building new sub-generation and switching stations Each station has similar PERT-CPM chart. A critical activity is obtaining planning building permission Legal team has queue of all projects in process. Gideon Weiss, University of Haifa, Scheduling and Queues, © 2008 12

Queueing vs Manufacturing: Single server queue Machine with controlled input A tandem of queues Queueing vs Manufacturing: Single server queue Machine with controlled input A tandem of queues Gideon Weiss, University of Haifa, Scheduling and Queues, © 2008 13

Multi Class Queueing Ntworks MCQN with Infinite Virtual Queues IVQ Queues/Classes 6 Initial Queue Multi Class Queueing Ntworks MCQN with Infinite Virtual Queues IVQ Queues/Classes 6 Initial Queue Levels 1 Routing Processes 2 3 Resources 5 Processing Durations 4 Network Dynamics Resource Allocation (Scheduling) Nominal Productio n Rates Gideon Weiss, University of Haifa, Scheduling and Queues, © 2008 14

IVQ’s Make Controlled Queueing Network even more interesting… What does a “good” control achieve? IVQ’s Make Controlled Queueing Network even more interesting… What does a “good” control achieve? Some Resource Sta b Qu le and eue L Siz ow es PUSH The Network PULL Hig ha n Th d Ba rou l gh anced pu Lo t wv dep aria n art ure ce of pro the ces s n tio a iliz rces Ut ou igh es H f. R o To Push Or To Pull? That is the question… Fluid oriented Approach: Choose a “good” nominal production rate (α)… Gideon Weiss, University of Haifa, Scheduling and Queues, © 2008 15

Finite horizon control of queueing networks Control MCQN (Q(t), T(t)), over 0<t<T objective Fluid Finite horizon control of queueing networks Control MCQN (Q(t), T(t)), over 0

A two machine 3 steps re-entrant line Minimum Makespan 360 Last Buffer First Served A two machine 3 steps re-entrant line Minimum Makespan 360 Last Buffer First Served 382 Gideon Weiss, University of Haifa, Scheduling and Queues, © 2008 Optimal Fluid Solution 352 17

Using Max Pressure to track fluid solution by MCQN-IVQ Gideon Weiss, University of Haifa, Using Max Pressure to track fluid solution by MCQN-IVQ Gideon Weiss, University of Haifa, Scheduling and Queues, © 2008 18

The KSRS network and the Push Pull network Kumar Seidman Rybko Stolyar network Push The KSRS network and the Push Pull network Kumar Seidman Rybko Stolyar network Push pull network (Kopzon, Nazarathy, W) Gideon Weiss, University of Haifa, Scheduling and Queues, © 2008 congested in heavy traffic Full utilization and stable queues 19

Summary • Queueing people should think BATCH (finite horizon) • Schedulers should think STREAM Summary • Queueing people should think BATCH (finite horizon) • Schedulers should think STREAM (queue) • A FLUID MODEL provides a tractable optimization problem • Tracking the fluid via MCQN w IVQ is ASYMPTOTICALLY OPTIMAL • It is all a question of SCALE • To verify your scale SIMULATE Gideon Weiss, University of Haifa, Scheduling and Queues, © 2008 20

Gideon Weiss, University of Haifa, Scheduling and Queues, © 2008 21 Gideon Weiss, University of Haifa, Scheduling and Queues, © 2008 21