Download - Tales of Time Scales
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Tales of Time Scales
Ward WhittAT&T Labs – Research
Florham Park, NJ
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New Book
Stochastic-Process LimitsAn Introduction to Stochastic-Process Limits
and Their Application to Queues
Springer 2001
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“I won’t waste a minute to read that book.”
- Felix Pollaczek
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“Ideal for anyone working on their third Ph.D. in queueing theory”
- Agner Krarup Erlang
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Flow Modification
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Flow Modification
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Multiple Classes
Single Server
1
2
m
m
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• Whitt, W. (1988) A light-traffic approximation for single-class departure processes from multi-class queues. Management Science 34, 1333-1346.
• Wischik, D. (1999) The output of a switch, or, effective bandwidths for networks. Queueing Systems 32, 383-396.
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M/G/1 Queue in a Random Environment
Let the arrival rate be a stochastic process with two states.
MMPP/G/1 is a special case.
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environment state
1 2
mean holding time (in environment state)
arrival rate
mean service time
Overall Traffic Intensity = 0.57
1 5
0.92 0.50
1 1
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environment state
1 2
mean holding time (in environment state)
arrival rate
mean service time
Overall Traffic Intensity = 0.75
1 5
2.00 0.50
1 1
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A Slowly Changing Environment
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environment state
1 2
mean holding time (in environment state)
arrival rate
mean service time
Overall Traffic Intensity = 0.57
1M 5M
0.92 0.50
1 1
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What Matters?• Environment Process?
• Arrival and Service Processes?
• M?
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Nearly Completely Decomposable Markov Chains
P. J. Courtois, Decomposability, 1977
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What if the queue is unstable in one of the environment states?
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environment state
1 2
mean holding time (in environment state)
arrival rate
mean service time
Overall Traffic Intensity = 0.75
1M 5M
2.00 0.50
1 1
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What Matters?• Environment Process?
• Arrival and Service Processes?
• M?
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Change Time Units
Measure time in units of M
i.e., divide time by M
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environment state
1 2
mean holding time (in environment state)
arrival rate
mean service time
Overall Traffic Intensity = 0.75
1 5
2.00M 0.50M
1/M 1/M
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Workload in Remaining Service TimeWith Deterministic Holding Times
1
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Steady-state workload tail probabilities in the MMPP/G/1 queue
0.40260 0.13739 0.04695 0.016040.41100 0.14350 0.05040 0.017700.44246 0.16168 0.06119 0.023160.52216 0.23002 0.01087 0.051680.37376 0.11418 0.03488 0.010660.37383 0.11425 0.03492 0.010670.37670 0.11669 0.03614 0.011200.38466 0.12398 0.03997 0.012890.37075 0.11183 0.03373 0.010170.37082 0.11189 0.03376 0.010190.37105 0.11208 0.03385 0.010230.37186 0.11281 0.03422 0.010380.37044 0.11159 0.03362 0.010130.37045 0.11160 0.03362 0.010130.37047 0.11164 0.03363 0.010130.37055 0.11169 0.03366 0.01015
4
22 , 4b
DEM
H c
4
22 , 4b
DEM
H c
4
22 , 4b
DEM
H c
4
22 , 4b
DEM
H c
2
3
4
10
10
10
10
sizefactor
M
service-timedistribution tail probability ( )P W x
0.5x 2.5x 4.5x 6.5x
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G. L. Choudhury, A. Mandelbaum, M. I. Reiman and W. Whitt, “Fluid and diffusion limits for queues in slowly changing environments.” Stochastic Models 13 (1997) 121-146.
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Thesis:
Heavy-traffic limits for queues can help expose phenomena occurring
at different time scales.
Asymptotically, there may be a separation of time scales.
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Network Status Probe
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Heavy-Traffic Perspective
Snapshot Principle
n-H Wn(nt) W(t)
0 < H < 1
n = (1 - )-1/(1-H)
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Server Scheduling
Multiple Classes
Single Server
With Delay and Switching Costs
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Heavy-Traffic Limit for Workload
Wn W
Wn(t) = n-H Wn(nt)
0 < H < 1
n = (1 - )-1/(1-H)
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One Approach: Polling
Multiple Classes
Single Server
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Heavy-Traffic Averaging Principle
h-1 f(Wi,n(t)) dt h-1 ( f(aiuW(t)) du ) dt
Wi,n(t) = n-H Wi,n(nt)
hs
s1
0hs
s
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• Coffman, E. G., Jr., Puhalskii, A. A. and Reiman, M. I. (1995) Polling systems with zero switchover times: a heavy-traffic averaging principle. Ann. Appl. Prob. 5, 681-719.
• Markowitz, D. M. and Wein, L. M. (2001) Heavy-traffic analysis of dynamic cyclic policies: a unified treatment of the single machine scheduling problem. Operations Res. 49, 246-270.
• Kushner, H. J. (2001) Heavy Traffic Analysis of Controlled Queueing and Communication Networks, Springer, New York.
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“My thesis has been that one path to the construction of a nontrivial theory of complex systems is by way of a
theory of hierarchy.” - H. A. Simon
•Holt, Modigliani, Muth and Simon, Planning Production, Inventories and Workforce, 1960.•Simon and Ando, Aggregation of variables in dynamic systems. Econometrica, 1961.•Ando, Fisher and Simon, Essays on the Structure of Social Science Models, 1963.
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Application to Manufacturing
MRP
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mrp MRPmaterial requirements
planningManufacturing Resources
Planning
Management information system
Expand bill of materials
Orlicky (1975)
Long-Range Planning (Strategic)Intermediate-Range Planning (Tactical)Short-Term Control (Operational)
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Hierarchical Decision Making in Stochastic Manufacturing Systems
- Suresh P. Sethi and Qing Zhang, 1994
0
inf [ ( ( )) ( ( ))]t
uE e h x t c u t dt
A
( ) ( ) ( ),
0 ( ) ( )
x t u t z t
u t K t