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Simple and explicit bounds formulti-server queues with universal 1

1−ρscaling

David A. Goldberg

Cornell

LNMB

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Punchline Model History Main results Proof Conclusion

Outline

1 Punchline

2 Model

3 History

4 Main results

5 Proof

6 Conclusion

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Punchline Model History Main results Proof Conclusion

Outline

1 Punchline

2 Model

3 History

4 Main results

5 Proof

6 Conclusion

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Punchline Model History Main results Proof Conclusion

11−ρ

11−ρ

Central insight of queueing theory:L (s.s num in q) scales as 1

1−ρ as ρ ↑ 1Many operational applications

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Punchline Model History Main results Proof Conclusion

11−ρ

11−ρ

Central insight of queueing theory:L (s.s num in q) scales as 1

1−ρ as ρ ↑ 1Many operational applications

beamer-tu-logo

Punchline Model History Main results Proof Conclusion

11−ρ

11−ρ

Central insight of queueing theory:L (s.s num in q) scales as 1

1−ρ as ρ ↑ 1Many operational applications

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Punchline Model History Main results Proof Conclusion

11−ρ?

11−ρ?

Only rigorously justified for G/G/n in a few special cases!Single serverExponential or deterministic service timesSpecial asymptotic regimes

Far less known is known when it comes to . . .Simple, explicit, non-asymptotic bounds

The exception is Kingman’s bound, but . . .Only for single server

A major difficulty is that any such bound . . .Must scale as 1

1−ρ even if n→∞ as ρ ↑ 1

Multi-server Kingman’s bound open for 50 years!

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Punchline Model History Main results Proof Conclusion

11−ρ?

11−ρ?

Only rigorously justified for G/G/n in a few special cases!Single serverExponential or deterministic service timesSpecial asymptotic regimes

Far less known is known when it comes to . . .Simple, explicit, non-asymptotic bounds

The exception is Kingman’s bound, but . . .Only for single server

A major difficulty is that any such bound . . .Must scale as 1

1−ρ even if n→∞ as ρ ↑ 1

Multi-server Kingman’s bound open for 50 years!

beamer-tu-logo

Punchline Model History Main results Proof Conclusion

11−ρ?

11−ρ?

Only rigorously justified for G/G/n in a few special cases!Single serverExponential or deterministic service timesSpecial asymptotic regimes

Far less known is known when it comes to . . .Simple, explicit, non-asymptotic bounds

The exception is Kingman’s bound, but . . .Only for single server

A major difficulty is that any such bound . . .Must scale as 1

1−ρ even if n→∞ as ρ ↑ 1

Multi-server Kingman’s bound open for 50 years!

beamer-tu-logo

Punchline Model History Main results Proof Conclusion

11−ρ?

11−ρ?

Only rigorously justified for G/G/n in a few special cases!Single serverExponential or deterministic service timesSpecial asymptotic regimes

Far less known is known when it comes to . . .Simple, explicit, non-asymptotic bounds

The exception is Kingman’s bound, but . . .Only for single server

A major difficulty is that any such bound . . .Must scale as 1

1−ρ even if n→∞ as ρ ↑ 1

Multi-server Kingman’s bound open for 50 years!

beamer-tu-logo

Punchline Model History Main results Proof Conclusion

11−ρ?

11−ρ?

Only rigorously justified for G/G/n in a few special cases!Single serverExponential or deterministic service timesSpecial asymptotic regimes

Far less known is known when it comes to . . .Simple, explicit, non-asymptotic bounds

The exception is Kingman’s bound, but . . .Only for single server

A major difficulty is that any such bound . . .Must scale as 1

1−ρ even if n→∞ as ρ ↑ 1

Multi-server Kingman’s bound open for 50 years!

beamer-tu-logo

Punchline Model History Main results Proof Conclusion

11−ρ?

11−ρ?

Only rigorously justified for G/G/n in a few special cases!Single serverExponential or deterministic service timesSpecial asymptotic regimes

Far less known is known when it comes to . . .Simple, explicit, non-asymptotic bounds

The exception is Kingman’s bound, but . . .Only for single server

A major difficulty is that any such bound . . .Must scale as 1

1−ρ even if n→∞ as ρ ↑ 1

Multi-server Kingman’s bound open for 50 years!

beamer-tu-logo

Punchline Model History Main results Proof Conclusion

11−ρ?

11−ρ?

Only rigorously justified for G/G/n in a few special cases!Single serverExponential or deterministic service timesSpecial asymptotic regimes

Far less known is known when it comes to . . .Simple, explicit, non-asymptotic bounds

The exception is Kingman’s bound, but . . .Only for single server

A major difficulty is that any such bound . . .Must scale as 1

1−ρ even if n→∞ as ρ ↑ 1

Multi-server Kingman’s bound open for 50 years!

beamer-tu-logo

Punchline Model History Main results Proof Conclusion

11−ρ?

11−ρ?

Only rigorously justified for G/G/n in a few special cases!Single serverExponential or deterministic service timesSpecial asymptotic regimes

Far less known is known when it comes to . . .Simple, explicit, non-asymptotic bounds

The exception is Kingman’s bound, but . . .Only for single server

A major difficulty is that any such bound . . .Must scale as 1

1−ρ even if n→∞ as ρ ↑ 1

Multi-server Kingman’s bound open for 50 years!

beamer-tu-logo

Punchline Model History Main results Proof Conclusion

11−ρ?

11−ρ?

Only rigorously justified for G/G/n in a few special cases!Single serverExponential or deterministic service timesSpecial asymptotic regimes

Far less known is known when it comes to . . .Simple, explicit, non-asymptotic bounds

The exception is Kingman’s bound, but . . .Only for single server

A major difficulty is that any such bound . . .Must scale as 1

1−ρ even if n→∞ as ρ ↑ 1

Multi-server Kingman’s bound open for 50 years!

beamer-tu-logo

Punchline Model History Main results Proof Conclusion

11−ρ?

11−ρ?

Only rigorously justified for G/G/n in a few special cases!Single serverExponential or deterministic service timesSpecial asymptotic regimes

Far less known is known when it comes to . . .Simple, explicit, non-asymptotic bounds

The exception is Kingman’s bound, but . . .Only for single server

A major difficulty is that any such bound . . .Must scale as 1

1−ρ even if n→∞ as ρ ↑ 1

Multi-server Kingman’s bound open for 50 years!

beamer-tu-logo

Punchline Model History Main results Proof Conclusion

11−ρ?

11−ρ?

Only rigorously justified for G/G/n in a few special cases!Single serverExponential or deterministic service timesSpecial asymptotic regimes

Far less known is known when it comes to . . .Simple, explicit, non-asymptotic bounds

The exception is Kingman’s bound, but . . .Only for single server

A major difficulty is that any such bound . . .Must scale as 1

1−ρ even if n→∞ as ρ ↑ 1

Multi-server Kingman’s bound open for 50 years!

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Punchline Model History Main results Proof Conclusion

11−ρ!

11−ρ !

Our main result resolves this open questionSimple and explicit bounds for E [L] scaling as 1

1−ρ

General G/G/n only requiring finite 2 + ε momentsHigher moments and tailsSteady-state probability of delayIn some cases we even beat 1

1−ρ

Implications for Halfin-Whitt regimeBroadly justifies the 1

1−ρ heuristic for multi-server queuesProof of concept , work to do!

beamer-tu-logo

Punchline Model History Main results Proof Conclusion

11−ρ!

11−ρ !

Our main result resolves this open questionSimple and explicit bounds for E [L] scaling as 1

1−ρ

General G/G/n only requiring finite 2 + ε momentsHigher moments and tailsSteady-state probability of delayIn some cases we even beat 1

1−ρ

Implications for Halfin-Whitt regimeBroadly justifies the 1

1−ρ heuristic for multi-server queuesProof of concept , work to do!

beamer-tu-logo

Punchline Model History Main results Proof Conclusion

11−ρ!

11−ρ !

Our main result resolves this open questionSimple and explicit bounds for E [L] scaling as 1

1−ρ

General G/G/n only requiring finite 2 + ε momentsHigher moments and tailsSteady-state probability of delayIn some cases we even beat 1

1−ρ

Implications for Halfin-Whitt regimeBroadly justifies the 1

1−ρ heuristic for multi-server queuesProof of concept , work to do!

beamer-tu-logo

Punchline Model History Main results Proof Conclusion

11−ρ!

11−ρ !

Our main result resolves this open questionSimple and explicit bounds for E [L] scaling as 1

1−ρ

General G/G/n only requiring finite 2 + ε momentsHigher moments and tailsSteady-state probability of delayIn some cases we even beat 1

1−ρ

Implications for Halfin-Whitt regimeBroadly justifies the 1

1−ρ heuristic for multi-server queuesProof of concept , work to do!

beamer-tu-logo

Punchline Model History Main results Proof Conclusion

11−ρ!

11−ρ !

Our main result resolves this open questionSimple and explicit bounds for E [L] scaling as 1

1−ρ

General G/G/n only requiring finite 2 + ε momentsHigher moments and tailsSteady-state probability of delayIn some cases we even beat 1

1−ρ

Implications for Halfin-Whitt regimeBroadly justifies the 1

1−ρ heuristic for multi-server queuesProof of concept , work to do!

beamer-tu-logo

Punchline Model History Main results Proof Conclusion

11−ρ!

11−ρ !

Our main result resolves this open questionSimple and explicit bounds for E [L] scaling as 1

1−ρ

General G/G/n only requiring finite 2 + ε momentsHigher moments and tailsSteady-state probability of delayIn some cases we even beat 1

1−ρ

Implications for Halfin-Whitt regimeBroadly justifies the 1

1−ρ heuristic for multi-server queuesProof of concept , work to do!

beamer-tu-logo

Punchline Model History Main results Proof Conclusion

11−ρ!

11−ρ !

Our main result resolves this open questionSimple and explicit bounds for E [L] scaling as 1

1−ρ

General G/G/n only requiring finite 2 + ε momentsHigher moments and tailsSteady-state probability of delayIn some cases we even beat 1

1−ρ

Implications for Halfin-Whitt regimeBroadly justifies the 1

1−ρ heuristic for multi-server queuesProof of concept , work to do!

beamer-tu-logo

Punchline Model History Main results Proof Conclusion

11−ρ!

11−ρ !

Our main result resolves this open questionSimple and explicit bounds for E [L] scaling as 1

1−ρ

General G/G/n only requiring finite 2 + ε momentsHigher moments and tailsSteady-state probability of delayIn some cases we even beat 1

1−ρ

Implications for Halfin-Whitt regimeBroadly justifies the 1

1−ρ heuristic for multi-server queuesProof of concept , work to do!

beamer-tu-logo

Punchline Model History Main results Proof Conclusion

11−ρ!

11−ρ !

Our main result resolves this open questionSimple and explicit bounds for E [L] scaling as 1

1−ρ

General G/G/n only requiring finite 2 + ε momentsHigher moments and tailsSteady-state probability of delayIn some cases we even beat 1

1−ρ

Implications for Halfin-Whitt regimeBroadly justifies the 1

1−ρ heuristic for multi-server queuesProof of concept , work to do!

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Punchline Model History Main results Proof Conclusion

Outline

1 Punchline

2 Model

3 History

4 Main results

5 Proof

6 Conclusion

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Punchline Model History Main results Proof Conclusion

FCFS GI/GI/n Queue

FCFS GI/GI/n Queue

Inter-arrival times i.i.d. ∼ AService times i.i.d. ∼ SµA = 1

E [A] , µS = 1E [S]

n serversTraffic intensity ρ = µA

nµS

Jobs served FCFSL: s.s. number waiting in queuePwait : s.s. prob. all servers busy

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Punchline Model History Main results Proof Conclusion

FCFS GI/GI/n Queue

FCFS GI/GI/n Queue

Inter-arrival times i.i.d. ∼ AService times i.i.d. ∼ SµA = 1

E [A] , µS = 1E [S]

n serversTraffic intensity ρ = µA

nµS

Jobs served FCFSL: s.s. number waiting in queuePwait : s.s. prob. all servers busy

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Punchline Model History Main results Proof Conclusion

FCFS GI/GI/n Queue

FCFS GI/GI/n Queue

Inter-arrival times i.i.d. ∼ AService times i.i.d. ∼ SµA = 1

E [A] , µS = 1E [S]

n serversTraffic intensity ρ = µA

nµS

Jobs served FCFSL: s.s. number waiting in queuePwait : s.s. prob. all servers busy

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Punchline Model History Main results Proof Conclusion

FCFS GI/GI/n Queue

FCFS GI/GI/n Queue

Inter-arrival times i.i.d. ∼ AService times i.i.d. ∼ SµA = 1

E [A] , µS = 1E [S]

n serversTraffic intensity ρ = µA

nµS

Jobs served FCFSL: s.s. number waiting in queuePwait : s.s. prob. all servers busy

beamer-tu-logo

Punchline Model History Main results Proof Conclusion

FCFS GI/GI/n Queue

FCFS GI/GI/n Queue

Inter-arrival times i.i.d. ∼ AService times i.i.d. ∼ SµA = 1

E [A] , µS = 1E [S]

n serversTraffic intensity ρ = µA

nµS

Jobs served FCFSL: s.s. number waiting in queuePwait : s.s. prob. all servers busy

beamer-tu-logo

Punchline Model History Main results Proof Conclusion

FCFS GI/GI/n Queue

FCFS GI/GI/n Queue

Inter-arrival times i.i.d. ∼ AService times i.i.d. ∼ SµA = 1

E [A] , µS = 1E [S]

n serversTraffic intensity ρ = µA

nµS

Jobs served FCFSL: s.s. number waiting in queuePwait : s.s. prob. all servers busy

beamer-tu-logo

Punchline Model History Main results Proof Conclusion

FCFS GI/GI/n Queue

FCFS GI/GI/n Queue

Inter-arrival times i.i.d. ∼ AService times i.i.d. ∼ SµA = 1

E [A] , µS = 1E [S]

n serversTraffic intensity ρ = µA

nµS

Jobs served FCFSL: s.s. number waiting in queuePwait : s.s. prob. all servers busy

beamer-tu-logo

Punchline Model History Main results Proof Conclusion

FCFS GI/GI/n Queue

FCFS GI/GI/n Queue

Inter-arrival times i.i.d. ∼ AService times i.i.d. ∼ SµA = 1

E [A] , µS = 1E [S]

n serversTraffic intensity ρ = µA

nµS

Jobs served FCFSL: s.s. number waiting in queuePwait : s.s. prob. all servers busy

beamer-tu-logo

Punchline Model History Main results Proof Conclusion

Outline

1 Punchline

2 Model

3 History

4 Main results

5 Proof

6 Conclusion

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Punchline Model History Main results Proof Conclusion

Early 20th Century

(a) Erlang

(b) Pollaczek (c) Khinchin

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Punchline Model History Main results Proof Conclusion

Early 20th Century

Early 20th Century

Model “invented” to study early telephone networksPioneering work by engineers such as ErlangSoon found many other applicationsErlang solves the M/M/n caseP-K formula for E[L] in M/G/1 case

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Punchline Model History Main results Proof Conclusion

Early 20th Century

Early 20th Century

Model “invented” to study early telephone networksPioneering work by engineers such as ErlangSoon found many other applicationsErlang solves the M/M/n caseP-K formula for E[L] in M/G/1 case

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Punchline Model History Main results Proof Conclusion

Early 20th Century

Early 20th Century

Model “invented” to study early telephone networksPioneering work by engineers such as ErlangSoon found many other applicationsErlang solves the M/M/n caseP-K formula for E[L] in M/G/1 case

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Punchline Model History Main results Proof Conclusion

Early 20th Century

Early 20th Century

Model “invented” to study early telephone networksPioneering work by engineers such as ErlangSoon found many other applicationsErlang solves the M/M/n caseP-K formula for E[L] in M/G/1 case

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Punchline Model History Main results Proof Conclusion

Early 20th Century

Early 20th Century

Model “invented” to study early telephone networksPioneering work by engineers such as ErlangSoon found many other applicationsErlang solves the M/M/n caseP-K formula for E[L] in M/G/1 case

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Punchline Model History Main results Proof Conclusion

Mid 20th Century

(d) Spitzer

(e) Lindley (f) Kendall

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Punchline Model History Main results Proof Conclusion

Mid 20th Century

Mid 20th Century

Great progress on single-server queueLindley’s recursionSpitzer’s identity

Little progress on multi-server queueKendall solves G/M/n casePollaczek: Extremely complicated transforms

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Punchline Model History Main results Proof Conclusion

Mid 20th Century

Mid 20th Century

Great progress on single-server queueLindley’s recursionSpitzer’s identity

Little progress on multi-server queueKendall solves G/M/n casePollaczek: Extremely complicated transforms

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Punchline Model History Main results Proof Conclusion

Mid 20th Century

Mid 20th Century

Great progress on single-server queueLindley’s recursionSpitzer’s identity

Little progress on multi-server queueKendall solves G/M/n casePollaczek: Extremely complicated transforms

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Punchline Model History Main results Proof Conclusion

Mid 20th Century

Mid 20th Century

Great progress on single-server queueLindley’s recursionSpitzer’s identity

Little progress on multi-server queueKendall solves G/M/n casePollaczek: Extremely complicated transforms

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Punchline Model History Main results Proof Conclusion

Mid 20th Century

Mid 20th Century

Great progress on single-server queueLindley’s recursionSpitzer’s identity

Little progress on multi-server queueKendall solves G/M/n casePollaczek: Extremely complicated transforms

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Punchline Model History Main results Proof Conclusion

Mid 20th Century

Mid 20th Century

Great progress on single-server queueLindley’s recursionSpitzer’s identity

Little progress on multi-server queueKendall solves G/M/n casePollaczek: Extremely complicated transforms

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Punchline Model History Main results Proof Conclusion

The 60’s

(g) Sir John Kingman

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Punchline Model History Main results Proof Conclusion

The 60’s

The 60’s

Kingman’s Bound for general G/G/1 queueE [L] ≤ 1

2

(Var [AµA] + ρ2Var [SµS]

)× 1

1−ρ

E [L] ≤ 12

(Var [AµA] + Var [SµS]

)× 1

1−ρ

Simple, explicit, general, scalableUseful in theory and practice12

(Var [AµA] + Var [SµS]

)is scale-free

Scales as 11−ρ as ρ ↑ 1 in a broad sense

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Punchline Model History Main results Proof Conclusion

The 60’s

The 60’s

Kingman’s Bound for general G/G/1 queueE [L] ≤ 1

2

(Var [AµA] + ρ2Var [SµS]

)× 1

1−ρ

E [L] ≤ 12

(Var [AµA] + Var [SµS]

)× 1

1−ρ

Simple, explicit, general, scalableUseful in theory and practice12

(Var [AµA] + Var [SµS]

)is scale-free

Scales as 11−ρ as ρ ↑ 1 in a broad sense

beamer-tu-logo

Punchline Model History Main results Proof Conclusion

The 60’s

The 60’s

Kingman’s Bound for general G/G/1 queueE [L] ≤ 1

2

(Var [AµA] + ρ2Var [SµS]

)× 1

1−ρ

E [L] ≤ 12

(Var [AµA] + Var [SµS]

)× 1

1−ρ

Simple, explicit, general, scalableUseful in theory and practice12

(Var [AµA] + Var [SµS]

)is scale-free

Scales as 11−ρ as ρ ↑ 1 in a broad sense

beamer-tu-logo

Punchline Model History Main results Proof Conclusion

The 60’s

The 60’s

Kingman’s Bound for general G/G/1 queueE [L] ≤ 1

2

(Var [AµA] + ρ2Var [SµS]

)× 1

1−ρ

E [L] ≤ 12

(Var [AµA] + Var [SµS]

)× 1

1−ρ

Simple, explicit, general, scalableUseful in theory and practice12

(Var [AµA] + Var [SµS]

)is scale-free

Scales as 11−ρ as ρ ↑ 1 in a broad sense

beamer-tu-logo

Punchline Model History Main results Proof Conclusion

The 60’s

The 60’s

Kingman’s Bound for general G/G/1 queueE [L] ≤ 1

2

(Var [AµA] + ρ2Var [SµS]

)× 1

1−ρ

E [L] ≤ 12

(Var [AµA] + Var [SµS]

)× 1

1−ρ

Simple, explicit, general, scalableUseful in theory and practice12

(Var [AµA] + Var [SµS]

)is scale-free

Scales as 11−ρ as ρ ↑ 1 in a broad sense

beamer-tu-logo

Punchline Model History Main results Proof Conclusion

The 60’s

The 60’s

Kingman’s Bound for general G/G/1 queueE [L] ≤ 1

2

(Var [AµA] + ρ2Var [SµS]

)× 1

1−ρ

E [L] ≤ 12

(Var [AµA] + Var [SµS]

)× 1

1−ρ

Simple, explicit, general, scalableUseful in theory and practice12

(Var [AµA] + Var [SµS]

)is scale-free

Scales as 11−ρ as ρ ↑ 1 in a broad sense

beamer-tu-logo

Punchline Model History Main results Proof Conclusion

The 60’s

The 60’s

Kingman’s Bound for general G/G/1 queueE [L] ≤ 1

2

(Var [AµA] + ρ2Var [SµS]

)× 1

1−ρ

E [L] ≤ 12

(Var [AµA] + Var [SµS]

)× 1

1−ρ

Simple, explicit, general, scalableUseful in theory and practice12

(Var [AµA] + Var [SµS]

)is scale-free

Scales as 11−ρ as ρ ↑ 1 in a broad sense

beamer-tu-logo

Punchline Model History Main results Proof Conclusion

The 60’s (cont.)

The 60’s (cont.)

Kingman’s heavy-traffic analysis for G/G/1 queueConsider a sequence of queues indexed by intensity ρLet Lρ be the s.s. r.v. for system with intensity ρ{(1− ρ)Lρ, ρ ↑ 1} ⇒ 1

2

(Var [AµA] + Var [SµS]

)× Expo(1)

Certain technical conditions requiredShows Kingman’s bound is tight as ρ ↑ 1

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Punchline Model History Main results Proof Conclusion

The 60’s (cont.)

The 60’s (cont.)

Kingman’s heavy-traffic analysis for G/G/1 queueConsider a sequence of queues indexed by intensity ρLet Lρ be the s.s. r.v. for system with intensity ρ{(1− ρ)Lρ, ρ ↑ 1} ⇒ 1

2

(Var [AµA] + Var [SµS]

)× Expo(1)

Certain technical conditions requiredShows Kingman’s bound is tight as ρ ↑ 1

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Punchline Model History Main results Proof Conclusion

The 60’s (cont.)

The 60’s (cont.)

Kingman’s heavy-traffic analysis for G/G/1 queueConsider a sequence of queues indexed by intensity ρLet Lρ be the s.s. r.v. for system with intensity ρ{(1− ρ)Lρ, ρ ↑ 1} ⇒ 1

2

(Var [AµA] + Var [SµS]

)× Expo(1)

Certain technical conditions requiredShows Kingman’s bound is tight as ρ ↑ 1

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Punchline Model History Main results Proof Conclusion

The 60’s (cont.)

The 60’s (cont.)

Kingman’s heavy-traffic analysis for G/G/1 queueConsider a sequence of queues indexed by intensity ρLet Lρ be the s.s. r.v. for system with intensity ρ{(1− ρ)Lρ, ρ ↑ 1} ⇒ 1

2

(Var [AµA] + Var [SµS]

)× Expo(1)

Certain technical conditions requiredShows Kingman’s bound is tight as ρ ↑ 1

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Punchline Model History Main results Proof Conclusion

The 60’s (cont.)

The 60’s (cont.)

Kingman’s heavy-traffic analysis for G/G/1 queueConsider a sequence of queues indexed by intensity ρLet Lρ be the s.s. r.v. for system with intensity ρ{(1− ρ)Lρ, ρ ↑ 1} ⇒ 1

2

(Var [AµA] + Var [SµS]

)× Expo(1)

Certain technical conditions requiredShows Kingman’s bound is tight as ρ ↑ 1

beamer-tu-logo

Punchline Model History Main results Proof Conclusion

The 60’s (cont.)

The 60’s (cont.)

Kingman’s heavy-traffic analysis for G/G/1 queueConsider a sequence of queues indexed by intensity ρLet Lρ be the s.s. r.v. for system with intensity ρ{(1− ρ)Lρ, ρ ↑ 1} ⇒ 1

2

(Var [AµA] + Var [SµS]

)× Expo(1)

Certain technical conditions requiredShows Kingman’s bound is tight as ρ ↑ 1

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Punchline Model History Main results Proof Conclusion

The 60’s (cont.)

The 60’s (cont.)

Kingman poses some open problemsMulti-server analogue of Kingman’s bound?Multi-server analogue of heavy-traffic analysis?

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Punchline Model History Main results Proof Conclusion

The 60’s (cont.)

The 60’s (cont.)

Kingman poses some open problemsMulti-server analogue of Kingman’s bound?Multi-server analogue of heavy-traffic analysis?

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Punchline Model History Main results Proof Conclusion

The 60’s (cont.)

The 60’s (cont.)

Kingman poses some open problemsMulti-server analogue of Kingman’s bound?Multi-server analogue of heavy-traffic analysis?

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Punchline Model History Main results Proof Conclusion

The 70’s

(h) Borovkov

(i) Whitt (j) Iglehart

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Punchline Model History Main results Proof Conclusion

The 70’s

The 70’s

Kollerstrom solves the multi-server heavy-traffic analysisFIXED n, ρ ↑ 1{(1− ρ)Lρ, ρ ↑ 1} ⇒ 1

2

(Var [AµA] + Var [SµS]

)× Expo(1)

System behaves like a single sped-up server as ρ ↑ 1Same 1

1−ρ scaling as single-server caseRelated results by Whitt, Borovkov, Iglehart, Loulou

Attempts to use for a multi-server Kingman’s boundComplicated corrections to the heavy-traffic approximationKollerstrom, Nagaev, KennedyAll require FIXED n, ρ ↑ 1No hope of general explicit bound

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Punchline Model History Main results Proof Conclusion

The 70’s

The 70’s

Kollerstrom solves the multi-server heavy-traffic analysisFIXED n, ρ ↑ 1{(1− ρ)Lρ, ρ ↑ 1} ⇒ 1

2

(Var [AµA] + Var [SµS]

)× Expo(1)

System behaves like a single sped-up server as ρ ↑ 1Same 1

1−ρ scaling as single-server caseRelated results by Whitt, Borovkov, Iglehart, Loulou

Attempts to use for a multi-server Kingman’s boundComplicated corrections to the heavy-traffic approximationKollerstrom, Nagaev, KennedyAll require FIXED n, ρ ↑ 1No hope of general explicit bound

beamer-tu-logo

Punchline Model History Main results Proof Conclusion

The 70’s

The 70’s

Kollerstrom solves the multi-server heavy-traffic analysisFIXED n, ρ ↑ 1{(1− ρ)Lρ, ρ ↑ 1} ⇒ 1

2

(Var [AµA] + Var [SµS]

)× Expo(1)

System behaves like a single sped-up server as ρ ↑ 1Same 1

1−ρ scaling as single-server caseRelated results by Whitt, Borovkov, Iglehart, Loulou

Attempts to use for a multi-server Kingman’s boundComplicated corrections to the heavy-traffic approximationKollerstrom, Nagaev, KennedyAll require FIXED n, ρ ↑ 1No hope of general explicit bound

beamer-tu-logo

Punchline Model History Main results Proof Conclusion

The 70’s

The 70’s

Kollerstrom solves the multi-server heavy-traffic analysisFIXED n, ρ ↑ 1{(1− ρ)Lρ, ρ ↑ 1} ⇒ 1

2

(Var [AµA] + Var [SµS]

)× Expo(1)

System behaves like a single sped-up server as ρ ↑ 1Same 1

1−ρ scaling as single-server caseRelated results by Whitt, Borovkov, Iglehart, Loulou

Attempts to use for a multi-server Kingman’s boundComplicated corrections to the heavy-traffic approximationKollerstrom, Nagaev, KennedyAll require FIXED n, ρ ↑ 1No hope of general explicit bound

beamer-tu-logo

Punchline Model History Main results Proof Conclusion

The 70’s

The 70’s

Kollerstrom solves the multi-server heavy-traffic analysisFIXED n, ρ ↑ 1{(1− ρ)Lρ, ρ ↑ 1} ⇒ 1

2

(Var [AµA] + Var [SµS]

)× Expo(1)

System behaves like a single sped-up server as ρ ↑ 1Same 1

1−ρ scaling as single-server caseRelated results by Whitt, Borovkov, Iglehart, Loulou

Attempts to use for a multi-server Kingman’s boundComplicated corrections to the heavy-traffic approximationKollerstrom, Nagaev, KennedyAll require FIXED n, ρ ↑ 1No hope of general explicit bound

beamer-tu-logo

Punchline Model History Main results Proof Conclusion

The 70’s

The 70’s

Kollerstrom solves the multi-server heavy-traffic analysisFIXED n, ρ ↑ 1{(1− ρ)Lρ, ρ ↑ 1} ⇒ 1

2

(Var [AµA] + Var [SµS]

)× Expo(1)

System behaves like a single sped-up server as ρ ↑ 1Same 1

1−ρ scaling as single-server caseRelated results by Whitt, Borovkov, Iglehart, Loulou

Attempts to use for a multi-server Kingman’s boundComplicated corrections to the heavy-traffic approximationKollerstrom, Nagaev, KennedyAll require FIXED n, ρ ↑ 1No hope of general explicit bound

beamer-tu-logo

Punchline Model History Main results Proof Conclusion

The 70’s

The 70’s

Kollerstrom solves the multi-server heavy-traffic analysisFIXED n, ρ ↑ 1{(1− ρ)Lρ, ρ ↑ 1} ⇒ 1

2

(Var [AµA] + Var [SµS]

)× Expo(1)

System behaves like a single sped-up server as ρ ↑ 1Same 1

1−ρ scaling as single-server caseRelated results by Whitt, Borovkov, Iglehart, Loulou

Attempts to use for a multi-server Kingman’s boundComplicated corrections to the heavy-traffic approximationKollerstrom, Nagaev, KennedyAll require FIXED n, ρ ↑ 1No hope of general explicit bound

beamer-tu-logo

Punchline Model History Main results Proof Conclusion

The 70’s

The 70’s

Kollerstrom solves the multi-server heavy-traffic analysisFIXED n, ρ ↑ 1{(1− ρ)Lρ, ρ ↑ 1} ⇒ 1

2

(Var [AµA] + Var [SµS]

)× Expo(1)

System behaves like a single sped-up server as ρ ↑ 1Same 1

1−ρ scaling as single-server caseRelated results by Whitt, Borovkov, Iglehart, Loulou

Attempts to use for a multi-server Kingman’s boundComplicated corrections to the heavy-traffic approximationKollerstrom, Nagaev, KennedyAll require FIXED n, ρ ↑ 1No hope of general explicit bound

beamer-tu-logo

Punchline Model History Main results Proof Conclusion

The 70’s

The 70’s

Kollerstrom solves the multi-server heavy-traffic analysisFIXED n, ρ ↑ 1{(1− ρ)Lρ, ρ ↑ 1} ⇒ 1

2

(Var [AµA] + Var [SµS]

)× Expo(1)

System behaves like a single sped-up server as ρ ↑ 1Same 1

1−ρ scaling as single-server caseRelated results by Whitt, Borovkov, Iglehart, Loulou

Attempts to use for a multi-server Kingman’s boundComplicated corrections to the heavy-traffic approximationKollerstrom, Nagaev, KennedyAll require FIXED n, ρ ↑ 1No hope of general explicit bound

beamer-tu-logo

Punchline Model History Main results Proof Conclusion

The 70’s

The 70’s

Kollerstrom solves the multi-server heavy-traffic analysisFIXED n, ρ ↑ 1{(1− ρ)Lρ, ρ ↑ 1} ⇒ 1

2

(Var [AµA] + Var [SµS]

)× Expo(1)

System behaves like a single sped-up server as ρ ↑ 1Same 1

1−ρ scaling as single-server caseRelated results by Whitt, Borovkov, Iglehart, Loulou

Attempts to use for a multi-server Kingman’s boundComplicated corrections to the heavy-traffic approximationKollerstrom, Nagaev, KennedyAll require FIXED n, ρ ↑ 1No hope of general explicit bound

beamer-tu-logo

Punchline Model History Main results Proof Conclusion

The 70’s

The 70’s

Kollerstrom solves the multi-server heavy-traffic analysisFIXED n, ρ ↑ 1{(1− ρ)Lρ, ρ ↑ 1} ⇒ 1

2

(Var [AµA] + Var [SµS]

)× Expo(1)

System behaves like a single sped-up server as ρ ↑ 1Same 1

1−ρ scaling as single-server caseRelated results by Whitt, Borovkov, Iglehart, Loulou

Attempts to use for a multi-server Kingman’s boundComplicated corrections to the heavy-traffic approximationKollerstrom, Nagaev, KennedyAll require FIXED n, ρ ↑ 1No hope of general explicit bound

beamer-tu-logo

Punchline Model History Main results Proof Conclusion

The 70’s (cont.)

The 70’s (cont.)

Kingman derives a simple bound for G/G/n queuesBy analyzing the cyclic routing boundE [L] ≤ 1

2

(Var [AµA] + n × Var [SµS]

)× 1

1−ρ

Later made rigorous by Wolff, Brumelle, MoriThe n in front of Var [SµS] renders it ineffective!

Especially when n→∞, ρ ↑ 1 together

Can we get rid of it?

beamer-tu-logo

Punchline Model History Main results Proof Conclusion

The 70’s (cont.)

The 70’s (cont.)

Kingman derives a simple bound for G/G/n queuesBy analyzing the cyclic routing boundE [L] ≤ 1

2

(Var [AµA] + n × Var [SµS]

)× 1

1−ρ

Later made rigorous by Wolff, Brumelle, MoriThe n in front of Var [SµS] renders it ineffective!

Especially when n→∞, ρ ↑ 1 together

Can we get rid of it?

beamer-tu-logo

Punchline Model History Main results Proof Conclusion

The 70’s (cont.)

The 70’s (cont.)

Kingman derives a simple bound for G/G/n queuesBy analyzing the cyclic routing boundE [L] ≤ 1

2

(Var [AµA] + n × Var [SµS]

)× 1

1−ρ

Later made rigorous by Wolff, Brumelle, MoriThe n in front of Var [SµS] renders it ineffective!

Especially when n→∞, ρ ↑ 1 together

Can we get rid of it?

beamer-tu-logo

Punchline Model History Main results Proof Conclusion

The 70’s (cont.)

The 70’s (cont.)

Kingman derives a simple bound for G/G/n queuesBy analyzing the cyclic routing boundE [L] ≤ 1

2

(Var [AµA] + n × Var [SµS]

)× 1

1−ρ

Later made rigorous by Wolff, Brumelle, MoriThe n in front of Var [SµS] renders it ineffective!

Especially when n→∞, ρ ↑ 1 together

Can we get rid of it?

beamer-tu-logo

Punchline Model History Main results Proof Conclusion

The 70’s (cont.)

The 70’s (cont.)

Kingman derives a simple bound for G/G/n queuesBy analyzing the cyclic routing boundE [L] ≤ 1

2

(Var [AµA] + n × Var [SµS]

)× 1

1−ρ

Later made rigorous by Wolff, Brumelle, MoriThe n in front of Var [SµS] renders it ineffective!

Especially when n→∞, ρ ↑ 1 together

Can we get rid of it?

beamer-tu-logo

Punchline Model History Main results Proof Conclusion

The 70’s (cont.)

The 70’s (cont.)

Kingman derives a simple bound for G/G/n queuesBy analyzing the cyclic routing boundE [L] ≤ 1

2

(Var [AµA] + n × Var [SµS]

)× 1

1−ρ

Later made rigorous by Wolff, Brumelle, MoriThe n in front of Var [SµS] renders it ineffective!

Especially when n→∞, ρ ↑ 1 together

Can we get rid of it?

beamer-tu-logo

Punchline Model History Main results Proof Conclusion

The 70’s (cont.)

The 70’s (cont.)

Kingman derives a simple bound for G/G/n queuesBy analyzing the cyclic routing boundE [L] ≤ 1

2

(Var [AµA] + n × Var [SµS]

)× 1

1−ρ

Later made rigorous by Wolff, Brumelle, MoriThe n in front of Var [SµS] renders it ineffective!

Especially when n→∞, ρ ↑ 1 together

Can we get rid of it?

beamer-tu-logo

Punchline Model History Main results Proof Conclusion

Digression: When n→∞, ρ ↑ 1 together

Halfin-Whitt regime

Halfin-Whitt scaling regime:Used to study quality-efficiency trade-off in service systemsMany servers, regular service times, sped-up arrivalsρ ∼ 1− Bn− 1

2

Pwait has a non-trivial limiting value as n→∞Introduced in 1981 by Halfin and WhittIntensely studied in 90’s and 2000’sProven that L scales (roughly) as 1

1−ρ ∼ n12

Complicated non-explicit measure-valued weak limitsHW,GM,PR,Reed,GG,DDG,AR,. . .

Heavy-traffic corrections and boundsDai, Brav., Gurvich, Leeuw., Zwart, Ramanan, . . .

No general, scalable, simple and explicit bounds

beamer-tu-logo

Punchline Model History Main results Proof Conclusion

Digression: When n→∞, ρ ↑ 1 together

Halfin-Whitt regime

Halfin-Whitt scaling regime:Used to study quality-efficiency trade-off in service systemsMany servers, regular service times, sped-up arrivalsρ ∼ 1− Bn− 1

2

Pwait has a non-trivial limiting value as n→∞Introduced in 1981 by Halfin and WhittIntensely studied in 90’s and 2000’sProven that L scales (roughly) as 1

1−ρ ∼ n12

Complicated non-explicit measure-valued weak limitsHW,GM,PR,Reed,GG,DDG,AR,. . .

Heavy-traffic corrections and boundsDai, Brav., Gurvich, Leeuw., Zwart, Ramanan, . . .

No general, scalable, simple and explicit bounds

beamer-tu-logo

Punchline Model History Main results Proof Conclusion

Digression: When n→∞, ρ ↑ 1 together

Halfin-Whitt regime

Halfin-Whitt scaling regime:Used to study quality-efficiency trade-off in service systemsMany servers, regular service times, sped-up arrivalsρ ∼ 1− Bn− 1

2

Pwait has a non-trivial limiting value as n→∞Introduced in 1981 by Halfin and WhittIntensely studied in 90’s and 2000’sProven that L scales (roughly) as 1

1−ρ ∼ n12

Complicated non-explicit measure-valued weak limitsHW,GM,PR,Reed,GG,DDG,AR,. . .

Heavy-traffic corrections and boundsDai, Brav., Gurvich, Leeuw., Zwart, Ramanan, . . .

No general, scalable, simple and explicit bounds

beamer-tu-logo

Punchline Model History Main results Proof Conclusion

Digression: When n→∞, ρ ↑ 1 together

Halfin-Whitt regime

Halfin-Whitt scaling regime:Used to study quality-efficiency trade-off in service systemsMany servers, regular service times, sped-up arrivalsρ ∼ 1− Bn− 1

2

Pwait has a non-trivial limiting value as n→∞Introduced in 1981 by Halfin and WhittIntensely studied in 90’s and 2000’sProven that L scales (roughly) as 1

1−ρ ∼ n12

Complicated non-explicit measure-valued weak limitsHW,GM,PR,Reed,GG,DDG,AR,. . .

Heavy-traffic corrections and boundsDai, Brav., Gurvich, Leeuw., Zwart, Ramanan, . . .

No general, scalable, simple and explicit bounds

beamer-tu-logo

Punchline Model History Main results Proof Conclusion

Digression: When n→∞, ρ ↑ 1 together

Halfin-Whitt regime

Halfin-Whitt scaling regime:Used to study quality-efficiency trade-off in service systemsMany servers, regular service times, sped-up arrivalsρ ∼ 1− Bn− 1

2

Pwait has a non-trivial limiting value as n→∞Introduced in 1981 by Halfin and WhittIntensely studied in 90’s and 2000’sProven that L scales (roughly) as 1

1−ρ ∼ n12

Complicated non-explicit measure-valued weak limitsHW,GM,PR,Reed,GG,DDG,AR,. . .

Heavy-traffic corrections and boundsDai, Brav., Gurvich, Leeuw., Zwart, Ramanan, . . .

No general, scalable, simple and explicit bounds

beamer-tu-logo

Punchline Model History Main results Proof Conclusion

Digression: When n→∞, ρ ↑ 1 together

Halfin-Whitt regime

Halfin-Whitt scaling regime:Used to study quality-efficiency trade-off in service systemsMany servers, regular service times, sped-up arrivalsρ ∼ 1− Bn− 1

2

Pwait has a non-trivial limiting value as n→∞Introduced in 1981 by Halfin and WhittIntensely studied in 90’s and 2000’sProven that L scales (roughly) as 1

1−ρ ∼ n12

Complicated non-explicit measure-valued weak limitsHW,GM,PR,Reed,GG,DDG,AR,. . .

Heavy-traffic corrections and boundsDai, Brav., Gurvich, Leeuw., Zwart, Ramanan, . . .

No general, scalable, simple and explicit bounds

beamer-tu-logo

Punchline Model History Main results Proof Conclusion

Digression: When n→∞, ρ ↑ 1 together

Halfin-Whitt regime

Halfin-Whitt scaling regime:Used to study quality-efficiency trade-off in service systemsMany servers, regular service times, sped-up arrivalsρ ∼ 1− Bn− 1

2

Pwait has a non-trivial limiting value as n→∞Introduced in 1981 by Halfin and WhittIntensely studied in 90’s and 2000’sProven that L scales (roughly) as 1

1−ρ ∼ n12

Complicated non-explicit measure-valued weak limitsHW,GM,PR,Reed,GG,DDG,AR,. . .

Heavy-traffic corrections and boundsDai, Brav., Gurvich, Leeuw., Zwart, Ramanan, . . .

No general, scalable, simple and explicit bounds

beamer-tu-logo

Punchline Model History Main results Proof Conclusion

Digression: When n→∞, ρ ↑ 1 together

Halfin-Whitt regime

Halfin-Whitt scaling regime:Used to study quality-efficiency trade-off in service systemsMany servers, regular service times, sped-up arrivalsρ ∼ 1− Bn− 1

2

Pwait has a non-trivial limiting value as n→∞Introduced in 1981 by Halfin and WhittIntensely studied in 90’s and 2000’sProven that L scales (roughly) as 1

1−ρ ∼ n12

Complicated non-explicit measure-valued weak limitsHW,GM,PR,Reed,GG,DDG,AR,. . .

Heavy-traffic corrections and boundsDai, Brav., Gurvich, Leeuw., Zwart, Ramanan, . . .

No general, scalable, simple and explicit bounds

beamer-tu-logo

Punchline Model History Main results Proof Conclusion

Digression: When n→∞, ρ ↑ 1 together

Halfin-Whitt regime

Halfin-Whitt scaling regime:Used to study quality-efficiency trade-off in service systemsMany servers, regular service times, sped-up arrivalsρ ∼ 1− Bn− 1

2

Pwait has a non-trivial limiting value as n→∞Introduced in 1981 by Halfin and WhittIntensely studied in 90’s and 2000’sProven that L scales (roughly) as 1

1−ρ ∼ n12

Complicated non-explicit measure-valued weak limitsHW,GM,PR,Reed,GG,DDG,AR,. . .

Heavy-traffic corrections and boundsDai, Brav., Gurvich, Leeuw., Zwart, Ramanan, . . .

No general, scalable, simple and explicit bounds

beamer-tu-logo

Punchline Model History Main results Proof Conclusion

Digression: When n→∞, ρ ↑ 1 together

Halfin-Whitt regime

Halfin-Whitt scaling regime:Used to study quality-efficiency trade-off in service systemsMany servers, regular service times, sped-up arrivalsρ ∼ 1− Bn− 1

2

Pwait has a non-trivial limiting value as n→∞Introduced in 1981 by Halfin and WhittIntensely studied in 90’s and 2000’sProven that L scales (roughly) as 1

1−ρ ∼ n12

Complicated non-explicit measure-valued weak limitsHW,GM,PR,Reed,GG,DDG,AR,. . .

Heavy-traffic corrections and boundsDai, Brav., Gurvich, Leeuw., Zwart, Ramanan, . . .

No general, scalable, simple and explicit bounds

beamer-tu-logo

Punchline Model History Main results Proof Conclusion

Digression: When n→∞, ρ ↑ 1 together

Halfin-Whitt regime

Halfin-Whitt scaling regime:Used to study quality-efficiency trade-off in service systemsMany servers, regular service times, sped-up arrivalsρ ∼ 1− Bn− 1

2

Pwait has a non-trivial limiting value as n→∞Introduced in 1981 by Halfin and WhittIntensely studied in 90’s and 2000’sProven that L scales (roughly) as 1

1−ρ ∼ n12

Complicated non-explicit measure-valued weak limitsHW,GM,PR,Reed,GG,DDG,AR,. . .

Heavy-traffic corrections and boundsDai, Brav., Gurvich, Leeuw., Zwart, Ramanan, . . .

No general, scalable, simple and explicit bounds

beamer-tu-logo

Punchline Model History Main results Proof Conclusion

Digression: When n→∞, ρ ↑ 1 together

Halfin-Whitt regime

Halfin-Whitt scaling regime:Used to study quality-efficiency trade-off in service systemsMany servers, regular service times, sped-up arrivalsρ ∼ 1− Bn− 1

2

Pwait has a non-trivial limiting value as n→∞Introduced in 1981 by Halfin and WhittIntensely studied in 90’s and 2000’sProven that L scales (roughly) as 1

1−ρ ∼ n12

Complicated non-explicit measure-valued weak limitsHW,GM,PR,Reed,GG,DDG,AR,. . .

Heavy-traffic corrections and boundsDai, Brav., Gurvich, Leeuw., Zwart, Ramanan, . . .

No general, scalable, simple and explicit bounds

beamer-tu-logo

Punchline Model History Main results Proof Conclusion

Digression: When n→∞, ρ ↑ 1 together

Halfin-Whitt regime

Halfin-Whitt scaling regime:Used to study quality-efficiency trade-off in service systemsMany servers, regular service times, sped-up arrivalsρ ∼ 1− Bn− 1

2

Pwait has a non-trivial limiting value as n→∞Introduced in 1981 by Halfin and WhittIntensely studied in 90’s and 2000’sProven that L scales (roughly) as 1

1−ρ ∼ n12

Complicated non-explicit measure-valued weak limitsHW,GM,PR,Reed,GG,DDG,AR,. . .

Heavy-traffic corrections and boundsDai, Brav., Gurvich, Leeuw., Zwart, Ramanan, . . .

No general, scalable, simple and explicit bounds

beamer-tu-logo

Punchline Model History Main results Proof Conclusion

Until the present

Until the present

In spite of a century of work on multi-server queues . . .No universal explicit 1

1−ρ boundsNo multi-server Kingman’s boundNormalized moments × 1

1−ρ

Daley has lamented / conjectured on this in 70’s,80’s,90’sSuch a bound may not even exist

Negative results of Gupta et al.Complexity of HW limits

beamer-tu-logo

Punchline Model History Main results Proof Conclusion

Until the present

Until the present

In spite of a century of work on multi-server queues . . .No universal explicit 1

1−ρ boundsNo multi-server Kingman’s boundNormalized moments × 1

1−ρ

Daley has lamented / conjectured on this in 70’s,80’s,90’sSuch a bound may not even exist

Negative results of Gupta et al.Complexity of HW limits

beamer-tu-logo

Punchline Model History Main results Proof Conclusion

Until the present

Until the present

In spite of a century of work on multi-server queues . . .No universal explicit 1

1−ρ boundsNo multi-server Kingman’s boundNormalized moments × 1

1−ρ

Daley has lamented / conjectured on this in 70’s,80’s,90’sSuch a bound may not even exist

Negative results of Gupta et al.Complexity of HW limits

beamer-tu-logo

Punchline Model History Main results Proof Conclusion

Until the present

Until the present

In spite of a century of work on multi-server queues . . .No universal explicit 1

1−ρ boundsNo multi-server Kingman’s boundNormalized moments × 1

1−ρ

Daley has lamented / conjectured on this in 70’s,80’s,90’sSuch a bound may not even exist

Negative results of Gupta et al.Complexity of HW limits

beamer-tu-logo

Punchline Model History Main results Proof Conclusion

Until the present

Until the present

In spite of a century of work on multi-server queues . . .No universal explicit 1

1−ρ boundsNo multi-server Kingman’s boundNormalized moments × 1

1−ρ

Daley has lamented / conjectured on this in 70’s,80’s,90’sSuch a bound may not even exist

Negative results of Gupta et al.Complexity of HW limits

beamer-tu-logo

Punchline Model History Main results Proof Conclusion

Until the present

Until the present

In spite of a century of work on multi-server queues . . .No universal explicit 1

1−ρ boundsNo multi-server Kingman’s boundNormalized moments × 1

1−ρ

Daley has lamented / conjectured on this in 70’s,80’s,90’sSuch a bound may not even exist

Negative results of Gupta et al.Complexity of HW limits

beamer-tu-logo

Punchline Model History Main results Proof Conclusion

Until the present

Until the present

In spite of a century of work on multi-server queues . . .No universal explicit 1

1−ρ boundsNo multi-server Kingman’s boundNormalized moments × 1

1−ρ

Daley has lamented / conjectured on this in 70’s,80’s,90’sSuch a bound may not even exist

Negative results of Gupta et al.Complexity of HW limits

beamer-tu-logo

Punchline Model History Main results Proof Conclusion

Until the present

Until the present

In spite of a century of work on multi-server queues . . .No universal explicit 1

1−ρ boundsNo multi-server Kingman’s boundNormalized moments × 1

1−ρ

Daley has lamented / conjectured on this in 70’s,80’s,90’sSuch a bound may not even exist

Negative results of Gupta et al.Complexity of HW limits

beamer-tu-logo

Punchline Model History Main results Proof Conclusion

Outline

1 Punchline

2 Model

3 History

4 Main results

5 Proof

6 Conclusion

beamer-tu-logo

Punchline Model History Main results Proof Conclusion

Multi-server Kingman’s Bound

Corollary

For any G/G/n queue s.t. E [A3],E [S3] <∞,

E [L] is at most

10500(

E[(SµS)

3]E[(AµA)3])3

× 11− ρ

.

beamer-tu-logo

Punchline Model History Main results Proof Conclusion

Multi-server Kingman’s Bound

Corollary

For any G/G/n queue s.t. E [A3],E [S3] <∞,

E [L] is at most

10500(

E[(SµS)

3]E[(AµA)3])3

× 11− ρ

.

beamer-tu-logo

Punchline Model History Main results Proof Conclusion

Universal and explicit Pwait bounds

Theorem

For any G/G/n queue s.t. E [A3],E [S3] <∞,

Pwait is at most

10500(

E[(SµS)

3]E[(AµA)3])3(

n(1− ρ)2)− 3

2.

“Kicks in” exactly in HW regime(n(1− ρ)2

)− 32= B−3

beamer-tu-logo

Punchline Model History Main results Proof Conclusion

Universal and explicit Pwait bounds

Theorem

For any G/G/n queue s.t. E [A3],E [S3] <∞,

Pwait is at most

10500(

E[(SµS)

3]E[(AµA)3])3(

n(1− ρ)2)− 3

2.

“Kicks in” exactly in HW regime(n(1− ρ)2

)− 32= B−3

beamer-tu-logo

Punchline Model History Main results Proof Conclusion

Universal and explicit Pwait bounds

Theorem

For any G/G/n queue s.t. E [A3],E [S3] <∞,

Pwait is at most

10500(

E[(SµS)

3]E[(AµA)3])3(

n(1− ρ)2)− 3

2.

“Kicks in” exactly in HW regime(n(1− ρ)2

)− 32= B−3

beamer-tu-logo

Punchline Model History Main results Proof Conclusion

Universal and explicit Pwait bounds

Theorem

For any G/G/n queue s.t. E [A3],E [S3] <∞,

Pwait is at most

10500(

E[(SµS)

3]E[(AµA)3])3(

n(1− ρ)2)− 3

2.

“Kicks in” exactly in HW regime(n(1− ρ)2

)− 32= B−3

beamer-tu-logo

Punchline Model History Main results Proof Conclusion

Better than 11−ρ

Corollary

For any G/G/n queue s.t. E [A3],E [S3] <∞,

E [L] is at most

10500 ×(

E[(SµS)

3]E[(AµA)3])3

×(

n(1− ρ)2)− 1

2 × 11− ρ

.

Vast generalization of M/M/n . . .E [L] = Pwait × ρ

1−ρ

beamer-tu-logo

Punchline Model History Main results Proof Conclusion

Better than 11−ρ

Corollary

For any G/G/n queue s.t. E [A3],E [S3] <∞,

E [L] is at most

10500 ×(

E[(SµS)

3]E[(AµA)3])3

×(

n(1− ρ)2)− 1

2 × 11− ρ

.

Vast generalization of M/M/n . . .E [L] = Pwait × ρ

1−ρ

beamer-tu-logo

Punchline Model History Main results Proof Conclusion

Better than 11−ρ

Corollary

For any G/G/n queue s.t. E [A3],E [S3] <∞,

E [L] is at most

10500 ×(

E[(SµS)

3]E[(AµA)3])3

×(

n(1− ρ)2)− 1

2 × 11− ρ

.

Vast generalization of M/M/n . . .E [L] = Pwait × ρ

1−ρ

beamer-tu-logo

Punchline Model History Main results Proof Conclusion

Better than 11−ρ

Corollary

For any G/G/n queue s.t. E [A3],E [S3] <∞,

E [L] is at most

10500 ×(

E[(SµS)

3]E[(AµA)3])3

×(

n(1− ρ)2)− 1

2 × 11− ρ

.

Vast generalization of M/M/n . . .E [L] = Pwait × ρ

1−ρ

beamer-tu-logo

Punchline Model History Main results Proof Conclusion

Other results in paper

More moments→ better boundsNeed at least 2 + ε

Explicit tail boundsImplications for H-W regime

beamer-tu-logo

Punchline Model History Main results Proof Conclusion

Other results in paper

More moments→ better boundsNeed at least 2 + ε

Explicit tail boundsImplications for H-W regime

beamer-tu-logo

Punchline Model History Main results Proof Conclusion

Other results in paper

More moments→ better boundsNeed at least 2 + ε

Explicit tail boundsImplications for H-W regime

beamer-tu-logo

Punchline Model History Main results Proof Conclusion

Other results in paper

More moments→ better boundsNeed at least 2 + ε

Explicit tail boundsImplications for H-W regime

beamer-tu-logo

Punchline Model History Main results Proof Conclusion

Outline

1 Punchline

2 Model

3 History

4 Main results

5 Proof

6 Conclusion

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Punchline Model History Main results Proof Conclusion

Proof Overview

Proof Overview

GG.13 bounds L by 1-D random walksupt≥0

(A(t)−

∑ni=1 Ni(t)

)G.16 similarly bounds Pwait

Previously analyzed asymptotically in HW regimeWe analyze universally and non-asymptoticallyMany novel explicit bounds . . .

Pooled renewal processesNegative drift r.w. with stat. inc.Maximal inequalities

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Punchline Model History Main results Proof Conclusion

Proof Overview

Proof Overview

GG.13 bounds L by 1-D random walksupt≥0

(A(t)−

∑ni=1 Ni(t)

)G.16 similarly bounds Pwait

Previously analyzed asymptotically in HW regimeWe analyze universally and non-asymptoticallyMany novel explicit bounds . . .

Pooled renewal processesNegative drift r.w. with stat. inc.Maximal inequalities

beamer-tu-logo

Punchline Model History Main results Proof Conclusion

Proof Overview

Proof Overview

GG.13 bounds L by 1-D random walksupt≥0

(A(t)−

∑ni=1 Ni(t)

)G.16 similarly bounds Pwait

Previously analyzed asymptotically in HW regimeWe analyze universally and non-asymptoticallyMany novel explicit bounds . . .

Pooled renewal processesNegative drift r.w. with stat. inc.Maximal inequalities

beamer-tu-logo

Punchline Model History Main results Proof Conclusion

Proof Overview

Proof Overview

GG.13 bounds L by 1-D random walksupt≥0

(A(t)−

∑ni=1 Ni(t)

)G.16 similarly bounds Pwait

Previously analyzed asymptotically in HW regimeWe analyze universally and non-asymptoticallyMany novel explicit bounds . . .

Pooled renewal processesNegative drift r.w. with stat. inc.Maximal inequalities

beamer-tu-logo

Punchline Model History Main results Proof Conclusion

Proof Overview

Proof Overview

GG.13 bounds L by 1-D random walksupt≥0

(A(t)−

∑ni=1 Ni(t)

)G.16 similarly bounds Pwait

Previously analyzed asymptotically in HW regimeWe analyze universally and non-asymptoticallyMany novel explicit bounds . . .

Pooled renewal processesNegative drift r.w. with stat. inc.Maximal inequalities

beamer-tu-logo

Punchline Model History Main results Proof Conclusion

Proof Overview

Proof Overview

GG.13 bounds L by 1-D random walksupt≥0

(A(t)−

∑ni=1 Ni(t)

)G.16 similarly bounds Pwait

Previously analyzed asymptotically in HW regimeWe analyze universally and non-asymptoticallyMany novel explicit bounds . . .

Pooled renewal processesNegative drift r.w. with stat. inc.Maximal inequalities

beamer-tu-logo

Punchline Model History Main results Proof Conclusion

Proof Overview

Proof Overview

GG.13 bounds L by 1-D random walksupt≥0

(A(t)−

∑ni=1 Ni(t)

)G.16 similarly bounds Pwait

Previously analyzed asymptotically in HW regimeWe analyze universally and non-asymptoticallyMany novel explicit bounds . . .

Pooled renewal processesNegative drift r.w. with stat. inc.Maximal inequalities

beamer-tu-logo

Punchline Model History Main results Proof Conclusion

Proof Overview

Proof Overview

GG.13 bounds L by 1-D random walksupt≥0

(A(t)−

∑ni=1 Ni(t)

)G.16 similarly bounds Pwait

Previously analyzed asymptotically in HW regimeWe analyze universally and non-asymptoticallyMany novel explicit bounds . . .

Pooled renewal processesNegative drift r.w. with stat. inc.Maximal inequalities

beamer-tu-logo

Punchline Model History Main results Proof Conclusion

Proof Overview

Proof Overview

GG.13 bounds L by 1-D random walksupt≥0

(A(t)−

∑ni=1 Ni(t)

)G.16 similarly bounds Pwait

Previously analyzed asymptotically in HW regimeWe analyze universally and non-asymptoticallyMany novel explicit bounds . . .

Pooled renewal processesNegative drift r.w. with stat. inc.Maximal inequalities

beamer-tu-logo

Punchline Model History Main results Proof Conclusion

Outline

1 Punchline

2 Model

3 History

4 Main results

5 Proof

6 Conclusion

beamer-tu-logo

Punchline Model History Main results Proof Conclusion

Summary

Summary

First multi-server analogue of Kingman’s boundExplicit bounds with universal 1

1−ρ scalingHigher moments and Pwait

Applications to HW regimeBroad theoretical foundation for 1

1−ρ

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Punchline Model History Main results Proof Conclusion

Summary

Summary

First multi-server analogue of Kingman’s boundExplicit bounds with universal 1

1−ρ scalingHigher moments and Pwait

Applications to HW regimeBroad theoretical foundation for 1

1−ρ

beamer-tu-logo

Punchline Model History Main results Proof Conclusion

Summary

Summary

First multi-server analogue of Kingman’s boundExplicit bounds with universal 1

1−ρ scalingHigher moments and Pwait

Applications to HW regimeBroad theoretical foundation for 1

1−ρ

beamer-tu-logo

Punchline Model History Main results Proof Conclusion

Summary

Summary

First multi-server analogue of Kingman’s boundExplicit bounds with universal 1

1−ρ scalingHigher moments and Pwait

Applications to HW regimeBroad theoretical foundation for 1

1−ρ

beamer-tu-logo

Punchline Model History Main results Proof Conclusion

Summary

Summary

First multi-server analogue of Kingman’s boundExplicit bounds with universal 1

1−ρ scalingHigher moments and Pwait

Applications to HW regimeBroad theoretical foundation for 1

1−ρ

beamer-tu-logo

Punchline Model History Main results Proof Conclusion

Future research

Future research

Get that constant down!Tighter analysisFundamentally different analysis

Bridge to known asymptotic results e.g. in HWBridge to moment results of Scheller-WolfHeavy tailsOther queueing modelsHow to trade off simplicity and accuracy in bounds?What do we want from our analyses?

beamer-tu-logo

Punchline Model History Main results Proof Conclusion

Future research

Future research

Get that constant down!Tighter analysisFundamentally different analysis

Bridge to known asymptotic results e.g. in HWBridge to moment results of Scheller-WolfHeavy tailsOther queueing modelsHow to trade off simplicity and accuracy in bounds?What do we want from our analyses?

beamer-tu-logo

Punchline Model History Main results Proof Conclusion

Future research

Future research

Get that constant down!Tighter analysisFundamentally different analysis

Bridge to known asymptotic results e.g. in HWBridge to moment results of Scheller-WolfHeavy tailsOther queueing modelsHow to trade off simplicity and accuracy in bounds?What do we want from our analyses?

beamer-tu-logo

Punchline Model History Main results Proof Conclusion

Future research

Future research

Get that constant down!Tighter analysisFundamentally different analysis

Bridge to known asymptotic results e.g. in HWBridge to moment results of Scheller-WolfHeavy tailsOther queueing modelsHow to trade off simplicity and accuracy in bounds?What do we want from our analyses?

beamer-tu-logo

Punchline Model History Main results Proof Conclusion

Future research

Future research

Get that constant down!Tighter analysisFundamentally different analysis

Bridge to known asymptotic results e.g. in HWBridge to moment results of Scheller-WolfHeavy tailsOther queueing modelsHow to trade off simplicity and accuracy in bounds?What do we want from our analyses?

beamer-tu-logo

Punchline Model History Main results Proof Conclusion

Future research

Future research

Get that constant down!Tighter analysisFundamentally different analysis

Bridge to known asymptotic results e.g. in HWBridge to moment results of Scheller-WolfHeavy tailsOther queueing modelsHow to trade off simplicity and accuracy in bounds?What do we want from our analyses?

beamer-tu-logo

Punchline Model History Main results Proof Conclusion

Future research

Future research

Get that constant down!Tighter analysisFundamentally different analysis

Bridge to known asymptotic results e.g. in HWBridge to moment results of Scheller-WolfHeavy tailsOther queueing modelsHow to trade off simplicity and accuracy in bounds?What do we want from our analyses?

beamer-tu-logo

Punchline Model History Main results Proof Conclusion

Future research

Future research

Get that constant down!Tighter analysisFundamentally different analysis

Bridge to known asymptotic results e.g. in HWBridge to moment results of Scheller-WolfHeavy tailsOther queueing modelsHow to trade off simplicity and accuracy in bounds?What do we want from our analyses?

beamer-tu-logo

Punchline Model History Main results Proof Conclusion

Future research

Future research

Get that constant down!Tighter analysisFundamentally different analysis

Bridge to known asymptotic results e.g. in HWBridge to moment results of Scheller-WolfHeavy tailsOther queueing modelsHow to trade off simplicity and accuracy in bounds?What do we want from our analyses?