e13e30aa9f9648145bc023e3d4edf84f.ppt

- Количество слайдов: 22

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

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 #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 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 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, 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 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: 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 -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

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 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 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 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? 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

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, Scheduling and Queues, © 2008 18

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 (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