C. G. Cassandras Division of Systems Engineering
and Dept. of Electrical and Computer Engineering and Center for Information and Systems Engineering
Boston University
Christos G. Cassandras CODES Lab. - Boston University
EVENT-DRIVEN CONTROL,
COMMUNICATION, AND OPTIMIZATION
TIME-DRIVEN v EVENT-DRIVEN CONTROL
Christos G. Cassandras CODES Lab. - Boston University
REFERENCE PLANT CONTROLLER
INPUT
-
+
SENSOR MEASURED OUTPUT
OUTPUT ERROR
REFERENCE PLANT CONTROLLER
INPUT
-
+
SENSOR MEASURED OUTPUT
OUTPUT ERROR
EVENT: g(STATE) ≤ 0
EVENT-DRIVEN CONTROL: Act only when needed (or on TIMEOUT) - not based on a clock
OUTLINE
Christos G. Cassandras CODES Lab. - Boston University
Reasons for EVENT-DRIVEN Control, Communication, and Optimization
EVENT-DRIVEN Control in Distributed Wireless Systems
EVENT-DRIVEN Sensitivity Analysis for Hybrid Systems
REASONS FOR EVENT-DRIVEN MODELS, CONTROL, OPTIMIZATION
Christos G. Cassandras CODES Lab. - Boston University
Many systems are naturally Discrete Event Systems (DES) (e.g., Internet) → all state transitions are event-driven Most of the rest are Hybrid Systems (HS) → some state transitions are event-driven Many systems are distributed → components interact asynchronously (through events) Time-driven sampling inherently inefficient (“open loop” sampling)
REASONS FOR EVENT-DRIVEN MODELS, CONTROL, OPTIMIZATION
Christos G. Cassandras CODES Lab. - Boston University
Many systems are stochastic → actions needed in response to random events Event-driven methods provide significant advantages in computation and estimation quality System performance is often more sensitive to event-driven components than to time-driven components
Many systems are wirelessly networked → energy constrained → time-driven communication consumes significant energy UNNECESSARILY!
CYBER-PHYSICAL SYSTEMS
Christos G. Cassandras CISE - CODES Lab. - Boston University
INTERNET
CYBER
PHYSICAL
Data collection: relatively easy…
Control: a challenge…
STATES
s1
s2 s3 s4
TIME t
TIME-DRIVEN SYSTEM
STATES
TIME t
STATE SPACE: X = ℜ
DYNAMICS: ( ) ,x f x t=
EVENT-DRIVEN SYSTEM
STATE SPACE: X s s s s= 1 2 3 4, , ,
DYNAMICS:
( )exfx ,'=t2
e2
x(t)
t3 t4 t5
e3 e4 e5 EVENTS
x(t)
t1
e1
Christos G. Cassandras CODES Lab. - Boston University
TIME-DRIVEN v EVENT-DRIVEN SYSTEMS
SYNCHRONOUS v ASYNCHRONOUS BEHAVIOR
Christos G. Cassandras CODES Lab. - Boston University
Wasted clock ticks
More wasted clock ticks
Even more wasted clock ticks …
INCREASING TIME GRANULARITY
Indistinguishable events
x + y x y
x y
TIME
Time-driven (synchronous) implementation: - Sum repeatedly evaluated unnecessarily - When evaluation is actually needed, it is done at the wrong times !
TIME
t1 t2
SYNCHRONOUS v ASYNCHRONOUS COMPUTATION
Christos G. Cassandras CODES Lab. - Boston University
SELECTED REFERENCES - EVENT-DRIVEN CONTROL, COMMUNICATION, ESTIMATION, OPTIMIZATION
Christos G. Cassandras CODES Lab. - Boston University
- Astrom, K.J., and B. M. Bernhardsson, “Comparison of Riemann and Lebesgue sampling for first order stochastic systems,” Proc. 41st Conf. Decision and Control, pp. 2011–2016, 2002. - T. Shima, S. Rasmussen, and P. Chandler, “UAV Team Decision and Control using Efficient Collaborative Estimation,” ASME J. of Dynamic Systems, Measurement, and Control, vol. 129, no. 5, pp. 609–619, 2007.
- Heemels, W. P. M. H., J. H. Sandee, and P. P. J. van den Bosch, “Analysis of event-driven controllers for linear systems,” Intl. J. Control, 81, pp. 571–590, 2008. - P. Tabuada, “Event-triggered real-time scheduling of stabilizing control tasks,” IEEE Trans. Autom. Control, vol. 52, pp. 1680–1685, 2007. - J. H. Sandee, W. P. M. H. Heemels, S. B. F. Hulsenboom, and P. P. J. van den Bosch, “Analysis and experimental validation of a sensor-based event-driven controller,” Proc. American Control Conf., pp. 2867–2874, 2007. - J. Lunze and D. Lehmann, “A state-feedback approach to event-based control,” Automatica, 46, pp. 211–215, 2010. - P. Wan and M. D. Lemmon, “Event triggered distributed optimization in sensor networks,” Proc. of 8th ACM/IEEE Intl. Conf. on Information Processing in Sensor Networks, 2009. - Zhong, M., and Cassandras, C.G., “Asynchronous Distributed Optimization with Event-Driven Communication”, IEEE Trans. on Automatic Control, AC-55, 12, pp. 2735-2750, 2010.
EVENT-DRIVEN CONTROL
IN DISTRIBUTED (USUALLY WIRELESS)
SYSTEMS
Christos G. Cassandras CODES Lab. - Boston University
MOTIVATIONAL PROBLEM: COVERAGE CONTROL
Deploy sensors to maximize “event” detection probability – unknown event locations – event sources may be mobile – sensors may be mobile
Perceived event density (data sources) over given region (mission space)
0
5
10
02
46
8100
10
20
30
40
50
Ω
R(x) (Hz/m2)
? ? ? ? ? ?
? ? ?
• Meguerdichian et al, IEEE INFOCOM, 2001 • Cortes et al, IEEE Trans. on Robotics and Automation, 2004 • Cassandras and Li, Eur. J. of Control, 2005 • Ganguli et al, American Control Conf., 2006 • Hussein and Stipanovic, American Control Conf., 2007 • Hokayem et al, American Control Conf., 2007
OPTIMAL COVERAGE IN A MAZE
Christos G. Cassandras CODES Lab. - Boston University
Zhong and Cassandras, 2008
http://www.bu.edu/codes/research/distributed-control/
COVERAGE: PROBLEM FORMULATION
Sensing attenuation: pi(x, si) monotonically decreasing in di(x) ≡ ||x - si||
Data source at x emits signal with energy E
N mobile sensors, each located at si∈R2
Signal observed by sensor node i (at si )
SENSING MODEL: ]),(|by Detected[),( iii sxAiPsxp ≡
( A(x) = data source emits at x )
Christos G. Cassandras CODES Lab. - Boston University
0
5
10
02
46
8100
10
20
30
40
50
Ω
R(x) (Hz/m2)
? ? ? ? ? ?
? ? ?
Joint detection prob. assuming sensor independence ( s = [s1,…,sN] : node locations)
[ ]∏=
−−=N
iii sxpxP
1
),(11),( s
OBJECTIVE: Determine locations s = [s1,…,sN] to maximize total Detection Probability:
),()(max dxxPxR∫Ω
ss
COVERAGE: PROBLEM FORMULATION
Christos G. Cassandras CODES Lab. - Boston University
Perceived event density
Event sensing probability
CONTINUED DISTRIBUTED COOPERATIVE SCHEME
Set
[ ] dxxpxRssHN
iiN ∫ ∏
Ω =
−−=1
1 )(11)(),,(
Christos G. Cassandras CODES Lab. - Boston University
Maximize H(s1,…,sN) by forcing nodes to move using gradient information:
[ ] dxxdxs
xdxpxpxR
sH
k
k
k
kN
kiii
k∫ ∏Ω ≠=
−∂∂
−=∂∂
)()()()(1)(
,1
ki
kki
ki s
Hss∂∂
+=+ β1Desired displacement = V·∆t
Cassandras and Li, EJC, 2005 Zhong and Cassandras, IEEE TAC, 2011
CONTINUED DISTRIBUTED COOPERATIVE SCHEME
… has to be autonomously evaluated by each node so as to determine how to move to next position:
CONTINUED
Use truncated pi(x) ⇒ Ω replaced by node neighborhood Ωi
Discretize pi(x) using a local grid
Christos G. Cassandras CODES Lab. - Boston University
[ ] dxxdxs
xdxpxpxR
sH
k
k
k
kN
kiii
k∫ ∏Ω ≠=
−∂∂
−=∂∂
)()()()(1)(
,1
ki
kki
ki s
Hss∂∂
+=+ β1
DISTRIBUTED COOPERATIVE OPTIMIZATION
Christos G. Cassandras CODES Lab. - Boston University
i
Nss
sts
ssHN
each on sconstraint ..
),,(min 1,,1
1
1
on sconstraint ..
),,(min1
sts
ssH Ns
N
Ns
sts
ssHN
on sconstraint ..
),,(min 1
…
N system components (processors, agents, vehicles, nodes), one common objective:
DISTRIBUTED COOPERATIVE OPTIMIZATION
Christos G. Cassandras CODES Lab. - Boston University
i
Controllable state si, i = 1,…,ni
))(()()1( kdksks iiii sα+=+
Step Size
Update Direction, usually ))(())(( kHkd ii ss −∇=
i
Ns
sts
ssHi
on sconstraint ..
),,(min 1
i requires knowledge of all s1,…,sN
Inter-node communication
SYNCHRONIZED (TIME-DRIVEN) COOPERATION
Christos G. Cassandras CODES Lab. - Boston University
1
2
3
COMMUNICATE + UPDATE
Drawbacks: Excessive communication (critical in wireless settings!) Faster nodes have to wait for slower ones Clock synchronization infeasible Bandwidth limitations Security risks
ASYNCHRONOUS COOPERATION
Christos G. Cassandras CODES Lab. - Boston University
1
2
3
Nodes not synchronized, delayed information used
Bertsekas and Tsitsiklis, 1997
Update frequency for each node is bounded + technical conditions
⇒ ))(()()1( kdksks iiii sα+=+
converges
ASYNCHRONOUS (EVENT-DRIVEN) COOPERATION
Christos G. Cassandras CODES Lab. - Boston University
2
3
UPDATE COMMUNICATE
UPDATE at i : locally determined, arbitrary (possibly periodic) COMMUNICATE from i : only when absolutely necessary
1
WHEN SHOULD A NODE COMMUNICATE?
Christos G. Cassandras CODES Lab. - Boston University
Node state at any time t : xi(t)
Node state at tk : si(k)
⇒ si(k) = xi(tk)
j
i tk
)(kjτEstimate examples:
))(()( kxks jj
ij τ= Most recent value
( )))(()())(()( kxdktkxks jjji
j
jkj
jij ταττ ⋅⋅
∆−
+= Linear prediction
: node j state estimated by node i )(ksijAT UPDATE TIME tk :
WHEN SHOULD A NODE COMMUNICATE?
Christos G. Cassandras CODES Lab. - Boston University
AT ANY TIME t :
If node i knows how j estimates its state, then it can evaluate )(tx ji
Node i uses • its own true state, xi(t) • the estimate that j uses, )(tx j
i
… and evaluates an ERROR FUNCTION ( ))(),( txtxg jii
Error Function examples: 21)()( ,)()( txtxtxtx j
iij
ii −−
: node i state estimated by node j )(tx ji
Christos G. Cassandras CODES Lab. - Boston University
i i
j
)(txi
δi
WHEN SHOULD A NODE COMMUNICATE?
Node i communicates its state to node j only when it detects that its true state xi(t) deviates from j’ estimate of it so that
)(tx ji
( ) ij
ii txtxg δ≥)(),(
( ))(),( txtxg jiiCompare ERROR FUNCTION to THRESHOLD δi
⇒ Event-Driven Control
Christos G. Cassandras CODES Lab. - Boston University
CONVERGENCE
Asynchronous distributed state update process at each i: ))(()()1( kdksks i
iii s⋅+=+ α Estimates of other nodes, evaluated by node i
THEOREM: Under certain conditions, there exist positive constants α and Kδ such that
0))((lim =∇∞→
kHk
s
INTERPRETATION: Event-driven cooperation achievable with minimal communication requirements ⇒ energy savings
Zhong and Cassandras, IEEE TAC, 2010
−=
otherwise)1(update sends if)(()(
kkkdKk
i
ii
i δδ δ s
Christos G. Cassandras CODES Lab. - Boston University
COONVERGENCE WHEN DELAYS ARE PRESENT
Red curve:
Black curve:
ij0τ t
0
( )kiδ
( )jii xxg ~,
( )jii xxg ,
ij3τij
1τ ij2τ ij
1σ ij2σ ij
3σ ij4σij
4τ
( )jii xxg ,
( )kiδ
0tij
3τij2τij
1τij0τ
Error function trajectory with NO DELAY
DELAY
Christos G. Cassandras CODES Lab. - Boston University
COONVERGENCE WHEN DELAYS ARE PRESENT
ASSUMPTION: There exists a non-negative integer D such that if a message is sent before tk-D from node i to node j, it will be received before tk. INTERPRETATION: at most D state update events can occur between a node sending a message and all destination nodes receiving this message.
Add a boundedness assumption:
THEOREM: Under certain conditions, there exist positive constants α and Kδ such that
0))((lim =∇∞→
kHk
s
NOTE: The requirements on α and Kδ depend on D and they are tighter.
Zhong and Cassandras, IEEE TAC, 2010
SYNCHRONOUS v ASYNCHRONOUS OPTIMAL COVERAGE PERFORMANCE
Christos G. Cassandras CODES Lab. - Boston University
SYNCHRONOUS v ASYNCHRONOUS: No. of communication events for a deployment problem with obstacles
SYNCHRONOUS v ASYNCHRONOUS: Achieving optimality in a problem with obstacles
Energy savings + Extended lifetime
DEMO: OPTIMAL DISTRIBUTED DEPLOYMENT WITH OBSTACLES – SIMULATED AND REAL
Christos G. Cassandras CODES Lab. - Boston University
Christos G. Cassandras CODES Lab. - Boston University
DEMO: REACTING TO EVENT DETECTION
Important to note:
There is no external control causing this behavior. Algorithm includes tracking functionality automatically
BOSTON UNIVERSITY TEST BEDS
SMARTS Kickoff Meeting Christos G. Cassandras CISE - CODES Lab. - Boston University
EVENT-DRIVEN SENSITIVITY
ANALYSIS
REAL-TIME STOCHASTIC OPTIMIZATION
CONTROL/DECISION (Parameterized by θ) SYSTEM PERFORMANCE
NOISE
Christos G. Cassandras CISE - CODES Lab. - Boston University
)]([ θLE
)]([max θθ
LEΘ∈
GOAL:
L(θ) GRADIENT ESTIMATOR
)(1 nnnn L θηθθ ∇+=+
)(θL∇ x(t)
DIFFICULTIES: - E[L(θ)] NOT available in closed form - not easy to evaluate - may not be a good estimate of
)(θL∇)(θL∇ )]([ θLE∇
REAL-TIME STOCHASTIC OPTIMIZATION FOR DES: INFINITESIMAL PERTURBATION ANALYSIS (IPA)
CONTROL/DECISION (Parameterized by θ)
Discrete Event System (DES)
PERFORMANCE
L(θ) IPA
NOISE
)(1 nnnn L θηθθ ∇+=+
Sample path
For many (but NOT all) DES: - Unbiased estimators - General distributions - Simple on-line implementation
[Ho and Cao, 1991], [Glasserman, 1991], [Cassandras, 1993, 2008]
)]([ θLE
Christos G. Cassandras CISE - CODES Lab. - Boston University
)(θL∇
x(t) Model
x(t)
REAL-TIME STOCHASTIC OPTIMIZATION: HYBRID SYSTEMS
CONTROL/DECISION (Parameterized by θ)
HYBRID SYSTEM
PERFORMANCE
L(θ) IPA )(1 nnnn L θηθθ ∇+=+
A general framework for an IPA theory in Hybrid Systems?
Sample path
Christos G. Cassandras CISE - CODES Lab. - Boston University
)]([ θLE
)(θL∇
NOISE
x(t)
Performance metric (objective function):
PERFORMANCE OPTIMIZATION AND IPA
( ) ( )[ ]TxLETxJ ),0,(;),0,(; θθθθ =
( ) ∑ ∫=
+
=N
kk
k
k
dttxLL0
1
),,(τ
τ
θθ
( ) ( ) ( )θθττ
θθ
∂∂
=′∂
∂=′ k
ktxtx ,,
NOTATION:
( )θθθ
dTxdJ ),0,(;
IPA goal: - Obtain unbiased estimates of , normally
- Then: θθηθθ
ddL n
nnn)(
1 +=+
( )θθ
ddL
Christos G. Cassandras CISE - CODES Lab. - Boston University
HYBRID AUTOMATA
Christos G. Cassandras CODES Lab. - Boston University
HYBRID AUTOMATA
Q: set of discrete states (modes) X: set of continuous states (normally Rn) E: set of events U: set of admissible controls
),,,,,,,,,,( 00 xqguardInvfUEXQGh ρφ=
f : vector field, φ : discrete state transition function,
XUXQf →××:QEXQ →××:φ
Inv: set defining an invariant condition (domain), guard: set defining a guard condition, ρ : reset function,
q0: initial discrete state x0: initial continuous state
XQInv ×⊆XQQguard ××⊆
XEXQQ →×××:ρ
HYBRID AUTOMATA
Christos G. Cassandras CODES Lab. - Boston University
HYBRID AUTOMATA Unreliable machine with timeouts
T=τ
00
0
==
τx 1
1
==
τ ux
][ T<τ
α
00
2
==
τx
β, Kx ≥
γ
0'0'
==
τx
0'0'
==
τx
IDLE BUSY DOWN
x(t) : physical state of part in machine τ(t): clock
α : START, β : STOP, γ : REPAIR
Invariant
Guard Reset
THE IPA CALCULUS
IPA: THREE FUNDAMENTAL EQUATIONS
1. Continuity at events:
Take d/dθ :
)()( −+ = kk xx ττ
kkkkkkk ffxx ')]()([)(')(' 1 τττττ +−−
−+ −+=
If no continuity, use reset condition ⇒ θ
δυρτd
xqqdx k),,,,()('
′=+
Christos G. Cassandras CISE - CODES Lab. - Boston University
System dynamics over (τk(θ), τk+1(θ)]: ),,( txfx k θ=
( ) ( ) ( )θθττ
θθ
∂∂
=′∂
∂=′ k
ktxtx ,,
NOTATION:
IPA: THREE FUNDAMENTAL EQUATIONS
Solve over (τk(θ), τk+1(θ)]: θ∂
∂+
∂∂
=)()(')()(' tftx
xtf
dttdx kk
′+∫
∂∂∫=′ ∫ +∂
∂−
∂∂ t
k
duxuf
kdu
xuf
k
v
k
kt
k
k
xdvevfetxτ
τθ
ττ )()()()()(
initial condition from 1 above
Christos G. Cassandras CISE - CODES Lab. - Boston University
2. Take d/dθ of system dynamics over (τk(θ), τk+1(θ)]: ),,( txfx k θ=
θ∂∂
+∂
∂=
)()(')()(' tftxxtf
dttdx kk
NOTE: If there are no events (pure time-driven system), IPA reduces to this equation
3. Get depending on the event type:
IPA: THREE FUNDAMENTAL EQUATIONS
kτ ′
- Exogenous event: By definition,
0=′kτ
0)),,(( =θτθ kk xg- Endogenous event: occurs when
′
∂∂
+∂∂
∂∂
−=′ −−
− )()(1
kkkk xxggf
xg τ
θττ
- Induced events:
)()( 1+
−
′
∂∂
−=′ kkkk
k yt
y τττ
Christos G. Cassandras CISE - CODES Lab. - Boston University
Ignoring resets and induced events:
IPA: THREE FUNDAMENTAL EQUATIONS
kkkkkkk ffxx ')]()([)(')(' 1 τττττ ⋅−+= +−−
−+
+∫
∂∂∫=′ ∫ +∂
∂−
∂∂ t
k
duxuf
kdu
xuf
k
v
k
kt
k
k
xdvevfetxτ
τθ
ττ )(')()()()(
0=′kτ
′
∂∂
+∂∂
∂∂
−=′ −−
− )()(1
kkkk xxggf
xg τ
θττor
1. 2.
3.
Christos G. Cassandras CISE - CODES Lab. - Boston University
2 1
3
)(' −kx τ
( ) ( )
( )θθττ
θθ
∂∂
=′
∂∂
=′
kk
txtx ,Recall:
Cassandras et al, Europ. J. Control, 2010
IPA PROPERTIES
Back to performance metric: ( ) ∑ ∫=
+
=N
kk
k
k
dttxLL0
1
),,(τ
τ
θθ
( ) ( )θ
θθ∂
∂=′ txLtxL k
k,,,,NOTATION:
Then: ( ) ∑ ∫=
++
′+⋅′−⋅′=
+N
kkkkkkkk
k
k
dttxLLLd
dL0
11
1
),,()()(τ
τ
θττττθθ
What happens at event times
What happens between event times
Christos G. Cassandras CISE - CODES Lab. - Boston University
IPA PROPERTIES: ROBUSTNESS
THEOREM 1: If either 1,2 holds, then dL(θ)/dθ depends only on information available at event times τk: 1. L(x,θ,t) is independent of t over [τk(θ), τk+1(θ)] for all k
2. L(x,θ,t) is only a function of x and for all t over [τk(θ), τk+1(θ)]:
0=∂∂
=∂∂
=∂∂
θkkk f
dtd
xf
dtd
xL
dtd
IMPLICATION: - Performance sensitivities can be obtained from information limited to event times, which is easily observed - No need to track system in between events !
( ) ∑ ∫=
++
′+⋅′−⋅′=
+N
kkkkkkkk
k
k
dttxLLLd
dL0
11
1
),,()()(τ
τ
θττττθθ
Christos G. Cassandras CISE - CODES Lab. - Boston University
[Yao and Cassandras, 2010]
IPA PROPERTIES : ROBUSTNESS
EXAMPLE WHERE THEOREM 1 APPLIES (simple tracking problem):
Christos G. Cassandras CISE - CODES Lab. - Boston University
k
k
k
kk
k
k
ddufa
xf
θθ=
∂∂
=∂∂
⇒ , Nk
twutxax
dtgtxE
kkkkkk
T
,,1 )()()( s.t.
)]()([min0
,
=++=
−∫
θ
φφθ 1 =
∂∂
⇒xL
NOTE: THEOREM 1 provides sufficient conditions only. IPA still depends on info. limited to event times if for “nice” functions uk(θk,t), e.g., bkθt
Nktwtutxax kkkkkk
,,1)(),()(
=++= θ
IPA PROPERTIES: DECOMPOSABILITY
THEOREM 2: Suppose an endogenous event occurs at τk with switching function g(x,θ). If , then is independent of fk−1.
If, in addition, then
IMPLICATION: Performance sensitivities are often reset to 0 ⇒ sample path can be conveniently decomposed
0)( =+kkf τ
0=θd
dg 0)( =′ +kx τ
)( +′ kx τ
Christos G. Cassandras CISE - CODES Lab. - Boston University
IPA PROPERTIES
EVENTS
Evaluating requires full knowledge of w and f values (obvious) );( θtx
However, may be independent of w and f values (NOT obvious) θ
θdtdx );(
It often depends only on: - event times τk - possibly )( 1
−+kf τ
);,,,( θtwuxfx =
τk τk+1
Christos G. Cassandras CISE - CODES Lab. - Boston University
IPA PROPERTIES
- No need for a detailed model (captured by fk) to describe state behavior in between events - This explains why simple abstractions of a complex stochastic system can be adequate to perform sensitivity analysis and optimization, as long as event times are accurately observed and local system behavior at these event times can also be measured. - This is true in abstractions of DES as HS since: Common performance metrics (e.g., workload) satisfy THEOREM 1
In many cases:
Christos G. Cassandras CISE - CODES Lab. - Boston University
THE CLASSIC SCHEDULING PROBLEM: cµ-RULE
Christos G. Cassandras CODES Lab. - Boston University
Problem:
cµ-rule: Always serve the non-empty queue with highest ciµi value
.,,1,0 , )(1min0
1,,1)(NicdttxcE
T i
T N
iiiNtu
=>
∫ ∑
=∈
x1(t)
x2(t)
xN(t)
µ1
µ2
µN
SERVICE RATES
ARRIVAL PROCESSES
… CONTROLLER:
,,1)( Ntu ∈
NOTE: cµ rule is an (almost) static control policy!
OPTIMALITY OF cµ-RULE
Christos G. Cassandras CODES Lab. - Boston University
• Deterministic model Smith, 1956
• Classical Queueing Theory: - M/G/1 system - Cox and Smith,1961 - Discrete time, general arrivals, geometrically distributed service - Baras et al., 1985; Buyukkoc et al., 1985 - Discrete time, service times with increasing/decreasing failure rates – Hirayama et al., 1989
• Fluid models: - Deterministic - Chen and Yao, 1993; Avram et al., 1995 - Fluid limits (heavy traffic) – Kingman, 1961; Whitt, 1968; Harrison, 1968; Mieghem, 1995
STOCHASTIC FLOW MODEL FOR SCHEDULING
Christos G. Cassandras CODES Lab. - Boston University
1);();(
2
2
1
1 ≤+µ
θµ
θ tutu
State dynamics:
−≥=
== + otherwise);()()()(,0)(0)();(
θαα
θtut
ttutxdt
tdxtfnn
nnnnn
>−
=
−=
0)())(1(
0)())(1(),(min)(
21
12
21
122
2
txtu
txtuttu
µµ
µµα
>=
=0)()(0)()(),(min
)(11
1111 txt
txtttu
θµθµα
]1 ,0[)( ∈tθ
);(1 θtx
);(2 θtx
)(1 tα
)(2 tα
1µ
2µ
);(1 θtu
);(2 θtu
Capacity Constraint:
nn tu µθ ≤≤ );(0
RANDOM PROCESSES
IPA FOR LINEAR HOLDING COSTS
Christos G. Cassandras CODES Lab. - Boston University
dttxctxcT
QT
)]()([1)( 220 11∫ +=θ Sample function:
NOTE: Result independent of inflow rate process αn(t) ⇒ Universality of cµ-rule !
THEOREM: If c1µ1 > c2µ2 , then 0)( <′ θQ
=
>= 0)()(
0)(1)(
11
1
1*
txttx
tµ
αθ cµ-rule is optimal
Kebarighotbi and Cassandras, J. DEDS, 2011
Proof: Use IPA CALCULUS to determine and show it is < 0 )(θQ′
CONCLUSIONS
Christos G. Cassandras CODES Lab. - Boston University
Seek to combine TIME-DRIVEN with EVENT-DRIVEN Control, Communication, and Optimization and exploit their relative advantages and disadvantages
EVENT-DRIVEN Control in Distributed Wireless Systems:
- Act only when necessary (when specific events occur)
EVENT-DRIVEN Sensitivity Analysis for Hybrid System
- Sensitivities depend mostly on events and are robust with respect to noise
THANK YOU
Top Related