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

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

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2 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 10x10x10 problem - demo of job shop simulator Fluid control of MCQN with IVQs joint work with Anat Kopzon and Yoni Nazarathy

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Gideon Weiss, University of Haifa, Scheduling and Queues, © Single Machine Scheduling - Batch 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

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

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Gideon Weiss, University of Haifa, Scheduling and Queues, © 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.

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

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

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

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

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

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Gideon Weiss, University of Haifa, Scheduling and Queues, © Job Shop Scheduling - Minimum Makespan, Fluid approach: Consider the famous 10x10 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.

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Gideon Weiss, University of Haifa, Scheduling and Queues, © 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 70s 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.

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

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

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Gideon Weiss, University of Haifa, Scheduling and Queues, © IVQ ’ s Make Controlled Queueing Network even more interesting … Some Resource The Network PUSH PULL To Push Or To Pull? That is the question… High Utilization of Resources High and Balanced Throughput Stable and Low Queue Sizes Low variance of the departure process What does a “good” control achieve? Fluid oriented Approach: Choose a “good” nominal production rate (α)…

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Gideon Weiss, University of Haifa, Scheduling and Queues, © Finite horizon control of queueing networks Fluid problem Control MCQN (Q(t),T(t)), over 0

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

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

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

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

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

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