[PPT] - Model and Motivation Many-server queue buffer N Many-server PowerPoint Presentation

SLIDE 1

FLUID MODELS OF MANY-SERVER QUEUES

WITH ABANDONMENT

Jiheng Zhang June 10, 2010

SLIDE 2

Background Stochastic Model Fluid Model Functional LLN Approximations

Model and Motivation

Many-server queue buffer N

Many-server Queues with Abandonment / Jiheng Zhang

SLIDE 3

Background Stochastic Model Fluid Model Functional LLN Approximations

Model and Motivation

Many-server queue buffer N Motivation: Customer call centers and other services areas.

Many-server Queues with Abandonment / Jiheng Zhang

SLIDE 4

Background Stochastic Model Fluid Model Functional LLN Approximations

Model and Motivation

Many-server queue with abandonment buffer N Motivation: Customer call centers and other services areas.

Many-server Queues with Abandonment / Jiheng Zhang

SLIDE 5

Background Stochastic Model Fluid Model Functional LLN Approximations

Many-server queue v.s. Single-server queue

Many-server Queues with Abandonment / Jiheng Zhang

SLIDE 6

Background Stochastic Model Fluid Model Functional LLN Approximations

Many-server queue v.s. Single-server queue

Large scale: high demand, need for high capacity Single server queue: increase speed Many-server queue: increase number of servers

Many-server Queues with Abandonment / Jiheng Zhang

SLIDE 7

Background Stochastic Model Fluid Model Functional LLN Approximations

A Real World Challenge

The service time is not exponentially distributed! Brown et. al. Statistical analysis of a telephone call center: a queueing-science perspective. JASA 2005 In this research Arrival process: general Service/patient time distribution: general

Many-server Queues with Abandonment / Jiheng Zhang

SLIDE 8

Background Stochastic Model Fluid Model Functional LLN Approximations

Literature Review

Many-server Queues Halfin and Whitt 1981 (M/M/N) Puhalskii and Reiman 2000 (G/Ph/N) Jelenkovi´ c, Mandelbaum and Momˇ cilovi´ c 2004 (G/D/N) Whitt 2005 (G/H∗

2/n/m)

Garmarnik and Momˇ cilovi´ c 2007 (G/La/N) Reed 2007, Puhalskii and Reed 2008 (G/G/N) Mandelbaum and Momˇ cilovi´ c 2008 (G/G/N) Kaspi and Ramanan 2009, Kaspi 2009 (G/G/N) . . . . . .

Many-server Queues with Abandonment / Jiheng Zhang

SLIDE 9

Background Stochastic Model Fluid Model Functional LLN Approximations

Literature Review

Many-server Queues with Abandonment Whitt 2004 (M/M/N + M) Zeltyn and Mandelbaum 2005 (G/M/N + G) Whitt 2006 (G/G/N + G) Puhalskii 2008 (Mt/Mt/Nt + Mt) Kang and Ramanan 2008 (G/G/N + G) Mandelbaum and Momˇ cilovi´ c 2009 (G/G/N + G) Dai, He and Tezcan 2009 (G/Ph/N + G) . . . . . .

Many-server Queues with Abandonment / Jiheng Zhang

SLIDE 10

Background Stochastic Model Fluid Model Functional LLN Approximations

System Dynamics – example with N = 2

3 t = 0

Many-server Queues with Abandonment / Jiheng Zhang

SLIDE 11

Background Stochastic Model Fluid Model Functional LLN Approximations

System Dynamics – example with N = 2

3 t = 0 2 5 arrival t = 1

Many-server Queues with Abandonment / Jiheng Zhang

SLIDE 12

Background Stochastic Model Fluid Model Functional LLN Approximations

System Dynamics – example with N = 2

3 t = 0 2 5 arrival t = 1 2 1 4 arrival t = 2

Many-server Queues with Abandonment / Jiheng Zhang

SLIDE 13

Background Stochastic Model Fluid Model Functional LLN Approximations

System Dynamics – example with N = 2

3 t = 0 2 5 arrival t = 1 2 1 4 arrival t = 2 2 3 departure t = 3

Many-server Queues with Abandonment / Jiheng Zhang

SLIDE 14

Background Stochastic Model Fluid Model Functional LLN Approximations

System Dynamics – example with N = 2

3 t = 0 2 5 arrival t = 1 2 1 4 arrival t = 2 2 3 departure t = 3 1 2 t = 4

Many-server Queues with Abandonment / Jiheng Zhang

SLIDE 15

Background Stochastic Model Fluid Model Functional LLN Approximations

System Dynamics – example with N = 2

3 t = 0 2 5 arrival t = 1 2 1 4 arrival t = 2 2 3 departure t = 3 1 2 t = 4 1 departure t = 5

Many-server Queues with Abandonment / Jiheng Zhang

SLIDE 16

Background Stochastic Model Fluid Model Functional LLN Approximations

Measure-valued State Descriptor

Server pool Z(t)(C): # of customers in server with remaining service time in C ⊂ (0, ∞)

Many-server Queues with Abandonment / Jiheng Zhang

SLIDE 17

Background Stochastic Model Fluid Model Functional LLN Approximations

Measure-valued State Descriptor

Server pool Z(t)(C): # of customers in server with remaining service time in C ⊂ (0, ∞) Evolution Z(t0 + t)(C) = Z(t0)(C + t) + . . . (t0, t0 + t) . . .

Many-server Queues with Abandonment / Jiheng Zhang

SLIDE 18

Background Stochastic Model Fluid Model Functional LLN Approximations

Measure-valued State Descriptor

Server pool Z(t)(C): # of customers in server with remaining service time in C ⊂ (0, ∞) Evolution Z(t0 + t)(C) = Z(t0)(C + t) + . . . (t0, t0 + t) . . . Richness Z(t) = Z(t)((0, ∞))

Many-server Queues with Abandonment / Jiheng Zhang

SLIDE 19

Background Stochastic Model Fluid Model Functional LLN Approximations

Measure-valued State Descriptor

Server pool Z(t)(C): # of customers in server with remaining service time in C ⊂ (0, ∞) Evolution Z(t0 + t)(C) = Z(t0)(C + t) + . . . (t0, t0 + t) . . . Richness Z(t) = Z(t)((0, ∞)) Literature

Gromoll, Puha & Williams ’02, Puha &Williams ’02, Gromoll ’06 Gromoll & Kurk ’07, Gromoll, Robert & Zwart ’08, . . . Zhang, Dai & Zwart ’07, ’08, Zhang & Zwart ’08

Many-server Queues with Abandonment / Jiheng Zhang

SLIDE 20

Background Stochastic Model Fluid Model Functional LLN Approximations

Measure-valued State Descriptor

Virtual buffer R(t)(C): # of customers in virtual buffer with remaining patient time in C ⊂(−∞, ∞)

Many-server Queues with Abandonment / Jiheng Zhang

SLIDE 21

Background Stochastic Model Fluid Model Functional LLN Approximations

Measure-valued State Descriptor

Virtual buffer R(t)(C): # of customers in virtual buffer with remaining patient time in C ⊂(−∞, ∞)

Many-server Queues with Abandonment / Jiheng Zhang

SLIDE 22

Background Stochastic Model Fluid Model Functional LLN Approximations

Measure-valued State Descriptor

Virtual buffer R(t)(C): # of customers in virtual buffer with remaining patient time in C ⊂(−∞, ∞) Evolution buffer 6 1 2 1 4

Many-server Queues with Abandonment / Jiheng Zhang

SLIDE 23

Background Stochastic Model Fluid Model Functional LLN Approximations

Measure-valued State Descriptor

Virtual buffer R(t)(C): # of customers in virtual buffer with remaining patient time in C ⊂(−∞, ∞) Evolution buffer 5 1 3

Many-server Queues with Abandonment / Jiheng Zhang

SLIDE 24

Background Stochastic Model Fluid Model Functional LLN Approximations

Measure-valued State Descriptor

Virtual buffer R(t)(C): # of customers in virtual buffer with remaining patient time in C ⊂(−∞, ∞) Evolution buffer 4

1
1

2

Many-server Queues with Abandonment / Jiheng Zhang

SLIDE 25

Background Stochastic Model Fluid Model Functional LLN Approximations

Measure-valued State Descriptor

Virtual buffer R(t)(C): # of customers in virtual buffer with remaining patient time in C ⊂(−∞, ∞) Evolution buffer 4

1
1

2 Richness Q(t) = R(t)((0, ∞)) R(t) = R(t)((−∞, ∞))

Many-server Queues with Abandonment / Jiheng Zhang

SLIDE 26

Background Stochastic Model Fluid Model Functional LLN Approximations

System Dynamics

Internal transfer process B(t) = E(t) − R(t)

Many-server Queues with Abandonment / Jiheng Zhang

SLIDE 27

Background Stochastic Model Fluid Model Functional LLN Approximations

System Dynamics

Internal transfer process B(t) = E(t) − R(t) 1 + B(t): index of the next customer to be served

Many-server Queues with Abandonment / Jiheng Zhang

SLIDE 28

Background Stochastic Model Fluid Model Functional LLN Approximations

System Dynamics

Internal transfer process B(t) = E(t) − R(t) 1 + B(t): index of the next customer to be served Stochastic dynamic equations R(t)(C) =

E(t)

i=1+B(t)

δui(C + t − ai), C ∈ B(R) Z(t)(C) = Z(0)(C + t) +

B(t)

i=1+B(0)

1{ui>τi−ai} δvi(C + t − τi), C ∈ B(R+)

Many-server Queues with Abandonment / Jiheng Zhang

SLIDE 29

Background Stochastic Model Fluid Model Functional LLN Approximations

System Dynamics

Internal transfer process B(t) = E(t) − R(t) 1 + B(t): index of the next customer to be served Stochastic dynamic equations

δui−(t−ai)C

R(t)(C) =

E(t)

i=1+B(t)

δui(C + t − ai), C ∈ B(R) Z(t)(C) = Z(0)(C + t) +

B(t)

i=1+B(0)

1{ui>τi−ai} δvi(C + t − τi), C ∈ B(R+)

Many-server Queues with Abandonment / Jiheng Zhang

SLIDE 30

Background Stochastic Model Fluid Model Functional LLN Approximations

System Dynamics

Internal transfer process B(t) = E(t) − R(t) 1 + B(t): index of the next customer to be served Stochastic dynamic equations

δui−(t−ai)C

R(t)(C) =

E(t)

i=1+B(t)

δui(C + t − ai), C ∈ B(R) Z(t)(C) = Z(0)(C + t) +

B(t)

i=1+B(0)

1{ui>τi−ai} δvi(C + t − τi), C ∈ B(R+)

Many-server Queues with Abandonment / Jiheng Zhang

SLIDE 31

Background Stochastic Model Fluid Model Functional LLN Approximations

System Dynamics

Internal transfer process B(t) = E(t) − R(t) 1 + B(t): index of the next customer to be served Stochastic dynamic equations

δui−(t−ai)C δvi−(t−τi)C

R(t)(C) =

E(t)

i=1+B(t)

δui(C + t − ai), C ∈ B(R) Z(t)(C) = Z(0)(C + t) +

B(t)

i=1+B(0)

1{ui>τi−ai} δvi(C + t − τi), C ∈ B(R+)

Many-server Queues with Abandonment / Jiheng Zhang

SLIDE 32

Background Stochastic Model Fluid Model Functional LLN Approximations

System Dynamics

Total number of customers X(t) = Q(t) + Z(t) Policy constraints Q(t) = (X(t) − N)+, Z(t) = (X(t) ∧ N)

Many-server Queues with Abandonment / Jiheng Zhang

SLIDE 33

Background Stochastic Model Fluid Model Functional LLN Approximations

Fluid Model

E(·): λ· {ui}: F (ϑF ∼ F) {vi}: G (ϑG ∼ G)

Many-server Queues with Abandonment / Jiheng Zhang

SLIDE 34

Background Stochastic Model Fluid Model Functional LLN Approximations

Fluid Model

E(·): λ· {ui}: F (ϑF ∼ F) {vi}: G (ϑG ∼ G) ¯ B(s) = λs − ¯ R(s)

Many-server Queues with Abandonment / Jiheng Zhang

SLIDE 35

Background Stochastic Model Fluid Model Functional LLN Approximations

Fluid Model

E(·): λ· {ui}: F (ϑF ∼ F) {vi}: G (ϑG ∼ G) ¯ B(s) = λs − ¯ R(s) Fluid dynamic equations ¯ R(t)(C) = t

t−

¯ R(t) λ

ϑF(C + t − s)dλs, C ∈ B(R) ¯ Z(t)(C) = ¯ Z(0)(C + t) + t ϑF( ¯ R(s) λ , ∞)ϑG(C + t − s)d¯ B(s), C ∈ B(R+)

Many-server Queues with Abandonment / Jiheng Zhang

SLIDE 36

Background Stochastic Model Fluid Model Functional LLN Approximations

Fluid Model

E(·): λ· {ui}: F (ϑF ∼ F) {vi}: G (ϑG ∼ G) ¯ B(s) = λs − ¯ R(s) has to be increasing! Fluid dynamic equations ¯ R(t)(C) = t

t−

¯ R(t) λ

ϑF(C + t − s)dλs, C ∈ B(R) ¯ Z(t)(C) = ¯ Z(0)(C + t) + t ϑF( ¯ R(s) λ , ∞)ϑG(C + t − s)d¯ B(s), C ∈ B(R+)

Many-server Queues with Abandonment / Jiheng Zhang

SLIDE 37

Background Stochastic Model Fluid Model Functional LLN Approximations

Fluid Model

E(·): λ· {ui}: F (ϑF ∼ F) {vi}: G (ϑG ∼ G) ¯ B(s) = λs − ¯ R(s) has to be increasing! Fluid dynamic equations ¯ R(t)(C) = t

t−

¯ R(t) λ

ϑF(C + t − s)dλs, C ∈ B(R) ¯ Z(t)(C) = ¯ Z(0)(C + t) + t ϑF( ¯ R(s) λ , ∞)ϑG(C + t − s)d¯ B(s), C ∈ B(R+) Constraints ¯ Q(t) = (¯ X(t) − N)+, ¯ Z(t) = (¯ X(t) ∧ N)

Many-server Queues with Abandonment / Jiheng Zhang

SLIDE 38

Background Stochastic Model Fluid Model Functional LLN Approximations

Existence and Uniqueness of Fluid Model Solution

Fluid model solution with initial condition ( ¯ R0, ¯ Z0) ( ¯ R(0), ¯ Z(0)) = ( ¯ R0, ¯ Z0) ( ¯ R(·), ¯ Z(·)) satisfies fluid dynamic equations and constraints

Many-server Queues with Abandonment / Jiheng Zhang

SLIDE 39

Background Stochastic Model Fluid Model Functional LLN Approximations

Existence and Uniqueness of Fluid Model Solution

Fluid model solution with initial condition ( ¯ R0, ¯ Z0) ( ¯ R(0), ¯ Z(0)) = ( ¯ R0, ¯ Z0) ( ¯ R(·), ¯ Z(·)) satisfies fluid dynamic equations and constraints THEOREM Assume that G is continuous, with 0 < µ < ∞, F(·) is Liptachitz continuous, or sup

x∈[0,∞)

hF(x) < ∞. There exists a unique solution to the fluid model (λ, F, G, N) for any valid initial condition ( ¯ R0, ¯ Z0).

Many-server Queues with Abandonment / Jiheng Zhang

SLIDE 40

Background Stochastic Model Fluid Model Functional LLN Approximations

Invariant State of Fluid Model

Invariant state ( ¯ R∞, ¯ Z∞) ( ¯ R(0), ¯ Z(0)) = ( ¯ R∞, ¯ Z∞) implies ( ¯ R(·), ¯ Z(·)) ≡ ( ¯ R∞, ¯ Z∞)

Many-server Queues with Abandonment / Jiheng Zhang

SLIDE 41

Background Stochastic Model Fluid Model Functional LLN Approximations

Invariant State of Fluid Model

Invariant state ( ¯ R∞, ¯ Z∞) ( ¯ R(0), ¯ Z(0)) = ( ¯ R∞, ¯ Z∞) implies ( ¯ R(·), ¯ Z(·)) ≡ ( ¯ R∞, ¯ Z∞) THEOREM The state ( ¯ R∞, ¯ Z∞) is an invariant state if and only if it satisfies ¯ R∞(Cx) = λ w Fc(x + s)ds, x ∈ R, ¯ Z∞(Cx) = min (ρ, 1) N[1 − Ge(x)], x ∈ R+, where w is a solution to the equation F(w) = max ρ − 1 ρ , 0

.

Many-server Queues with Abandonment / Jiheng Zhang

SLIDE 42

Background Stochastic Model Fluid Model Functional LLN Approximations

Fluid Model Analysis

Replace C by Cx = (x, ∞), (note that ϑF(Cx) = Fc(x)) ¯ R(t)(Cx) = λ t

t−

¯ R(t) λ

Fc(x + t − s)ds, x ∈ R, ¯ Z(t)(Cx) = ¯ Z(0)(Cx + t) + t Fc( ¯ R(s) λ )Gc(x + t − s)d¯ B(s), x ∈ R+,

Many-server Queues with Abandonment / Jiheng Zhang

SLIDE 43

Background Stochastic Model Fluid Model Functional LLN Approximations

Fluid Model Analysis

Replace C by Cx = (x, ∞), (note that ϑF(Cx) = Fc(x)) ¯ R(t)(Cx) = λ t

t−

¯ R(t) λ

Fc(x + t − s)ds, x ∈ R, ¯ Z(t)(Cx) = ¯ Z(0)(Cx + t) + t Fc( ¯ R(s) λ )Gc(x + t − s)d¯ B(s), x ∈ R+, The functional equation ¯ X(t) = ζ0(t) + ρ t H

(¯

X(t − s) − 1)+ dGe(s) + t (¯ X(t − s) − 1)+dG(s). where H(x) = Fc(F−1

e (α

λx)).

Many-server Queues with Abandonment / Jiheng Zhang

SLIDE 44

Background Stochastic Model Fluid Model Functional LLN Approximations

The Special Case with Exponential Distribution

Now, we specialize in the case with exponential distribution, i.e. F(t) = Fe(t) = 1 − e−αt, G(t) = Ge(t) = 1 − e−µt.

Many-server Queues with Abandonment / Jiheng Zhang

SLIDE 45

Background Stochastic Model Fluid Model Functional LLN Approximations

The Special Case with Exponential Distribution

Now, we specialize in the case with exponential distribution, i.e. F(t) = Fe(t) = 1 − e−αt, G(t) = Ge(t) = 1 − e−µt. Now the key equation becomes ¯ X(t) = ζ0(t) + ρ t

1 − α

λ

(¯

X(t − s) − 1)+ µe−µsds + t (¯ X(t − s) − 1)+µe−µsds, with ζ0(t) = ¯ X0e−µt.

Many-server Queues with Abandonment / Jiheng Zhang

SLIDE 46

Background Stochastic Model Fluid Model Functional LLN Approximations

The Special Case with Exponential Distribution

Now, we specialize in the case with exponential distribution, i.e. F(t) = Fe(t) = 1 − e−αt, G(t) = Ge(t) = 1 − e−µt. Now the key equation becomes ¯ X(t) = ζ0(t) + ρ t

1 − α

λ

(¯

X(t − s) − 1)+ µe−µsds + t (¯ X(t − s) − 1)+µe−µsds, with ζ0(t) = ¯ X0e−µt. After some algebra, we get ¯ X′(t) = µ(ρ − 1) − α(¯ X(t) − 1)+ + µ(¯ X(t) − 1)−. (Whitt 04)

Many-server Queues with Abandonment / Jiheng Zhang

SLIDE 47

Background Stochastic Model Fluid Model Functional LLN Approximations

Fluid Scaling and Limiting Regimes

A sequence of systems indexed by the number of servers n. Fluid scaling ¯ Rn(t) = 1 nRn(t), ¯ Zn(t) = 1 nZn(t),

Many-server Queues with Abandonment / Jiheng Zhang

SLIDE 48

Background Stochastic Model Fluid Model Functional LLN Approximations

Fluid Scaling and Limiting Regimes

A sequence of systems indexed by the number of servers n. Fluid scaling ¯ Rn(t) = 1 nRn(t), ¯ Zn(t) = 1 nZn(t), Arrival rate of the nth system λn ∼ nλ. ρn = λn nµn → ρ ∈ (0, ∞)

Many-server Queues with Abandonment / Jiheng Zhang

SLIDE 49

Background Stochastic Model Fluid Model Functional LLN Approximations

Fluid Scaling and Limiting Regimes

A sequence of systems indexed by the number of servers n. Fluid scaling ¯ Rn(t) = 1 nRn(t), ¯ Zn(t) = 1 nZn(t), Arrival rate of the nth system λn ∼ nλ. ρn = λn nµn → ρ ∈ (0, ∞)      > 1, ED = 1, QED < 1, QD

Many-server Queues with Abandonment / Jiheng Zhang

SLIDE 50

Background Stochastic Model Fluid Model Functional LLN Approximations

Fluid Scaling and Limiting Regimes

A sequence of systems indexed by the number of servers n. Fluid scaling ¯ Rn(t) = 1 nRn(t), ¯ Zn(t) = 1 nZn(t), Arrival rate of the nth system λn ∼ nλ. ρn = λn nµn → ρ ∈ (0, ∞)      > 1, ED = 1, QED < 1, QD Constraints ¯ Qn(t) = (¯ Xn(t) − 1)+, ¯ Zn(t) = (¯ Xn(t) ∧ 1)

Many-server Queues with Abandonment / Jiheng Zhang

SLIDE 51

Background Stochastic Model Fluid Model Functional LLN Approximations

Functional Law of Large Numbers

Assumption A:

1

¯ En(·) ⇒ λ·

2 ϑn

F → ϑF, ϑn G → ϑG

3 µn → µ 4 ( ¯

Rn(0), ¯ Zn(0)) ⇒ ( ¯ R0, ¯ Z0)

5

¯ R0 and ¯ Z0 has no atoms THEOREM Under assumption A ( ¯ Rn(·), ¯ Zn(·)) ⇒ ( ¯ R(·), ¯ Z(·)) as n → ∞, where ( ¯ R(·), ¯ Z(·)) is almost surely the fluid model solution to (λ, F, G, 1) with initial condition ( ¯ R0, ¯ Z0).

Many-server Queues with Abandonment / Jiheng Zhang

SLIDE 52

Background Stochastic Model Fluid Model Functional LLN Approximations

Performance Evaluation

Approximation formulas E(W|S) = w, F(w) = max ((ρ − 1)/ρ, 0) E(Q) = λ αFe(w)

Many-server Queues with Abandonment / Jiheng Zhang

SLIDE 53

Background Stochastic Model Fluid Model Functional LLN Approximations

Performance Evaluation

Approximation formulas E(W|S) = w, F(w) = max ((ρ − 1)/ρ, 0) E(Q) = λ αFe(w) M/GI/100-GI, λ = 120, µ = 1, α = 1 (Whitt 2006) Abd. Ser. E[Q] E[W|S] E2 E2 40.25 ± 0.057 0.353 ± 0.00051 LN(1, 4) 39.56 ± 0.097 0.343 ± 0.00094 Approximation 41.11 0.365 LN(1, 4) E2 14.51 ± 0.018 0.126 ± 0.00017 LN(1, 4) 14.52 ± 0.043 0.125 ± 0.00027 Approximation 14.63 0.131

Many-server Queues with Abandonment / Jiheng Zhang

SLIDE 54

Background Stochastic Model Fluid Model Functional LLN Approximations

A Missing Gap

Interchange of Steady State and Heavy Traffic Limits ( ¯ Rn(t), ¯ Zn(t)) ( ¯ Rn

∞, ¯

Zn

∞)

t → ∞

Many-server Queues with Abandonment / Jiheng Zhang

SLIDE 55

Background Stochastic Model Fluid Model Functional LLN Approximations

A Missing Gap

Interchange of Steady State and Heavy Traffic Limits ( ¯ Rn(t), ¯ Zn(t)) ( ¯ Rn

∞, ¯

Zn

∞)

t → ∞ ( ¯ R(t), ¯ Z(t)) n → ∞

Many-server Queues with Abandonment / Jiheng Zhang

SLIDE 56

Background Stochastic Model Fluid Model Functional LLN Approximations

A Missing Gap

Interchange of Steady State and Heavy Traffic Limits ( ¯ Rn(t), ¯ Zn(t)) ( ¯ Rn

∞, ¯

Zn

∞)

t → ∞ ( ¯ R(t), ¯ Z(t)) n → ∞ ( ¯ R∞, ¯ Z∞) t → ∞

Many-server Queues with Abandonment / Jiheng Zhang

SLIDE 57

Background Stochastic Model Fluid Model Functional LLN Approximations

A Missing Gap

Interchange of Steady State and Heavy Traffic Limits ( ¯ Rn(t), ¯ Zn(t)) ( ¯ Rn

∞, ¯

Zn

∞)

t → ∞ ( ¯ R(t), ¯ Z(t)) n → ∞ ( ¯ R∞, ¯ Z∞) t → ∞ n → ∞

Many-server Queues with Abandonment / Jiheng Zhang

SLIDE 58

Background Stochastic Model Fluid Model Functional LLN Approximations

Questions?

Many-server Queues with Abandonment / Jiheng Zhang

SLIDE 59

Background Stochastic Model Fluid Model Functional LLN Approximations

Thank you!

Many-server Queues with Abandonment / Jiheng Zhang