USING SIMULATION TO STUDY SERVICE-RATE CONTROLS TO …ww2040/wsc_Ni_120515.pdf · 2015-12-05 · We use simulation to study service-rate controls to stabilize performance in a single-server

USING SIMULATION TO STUDY SERVICE-RATE CONTROLSTO STABILIZE PERFORMANCE IN A SINGLE-SERVER QUEUE

WITH TIME-VARYING ARRIVAL RATE

Ni Ma and Ward Whitt

Columbia University

December 5, 2015

Ni Ma and Ward Whitt (CU) Stabilizing Performance December 5, 2015 1 / 31

Outline

1 MotivationStabilizing PerformanceService-Rate Controls

2 The Model

3 Simulation Methods For Nonstationary ModelsGenerating the Arrival ProcessGenerating the Service Times

4 Simulation Experiments

5 Simulation Results


Motivation


Motivation: Stabilizing Performance

Given the time-varying arrival rates, we are interested in an algorithmthat can stabilize performance of the queueing system, e.g.expected delay, delay probability, expected queue length.

Earlier papers that study server-staffing to stabilize performance inmulti-server queues with time-varying arrivals.

M. Defraeye and I. Van Niewenhuyse (2013) Controlling excessive waitingtimes in small service systems with time-varying demand: an extension of theISA algorithm. Decision Support Systems 54(4), 1558 – 1567.

Y. Liu and W. Whitt (2012) Stabilizing customer abandonment inmany-server queues with time-varying arrivals. Oper. Res. 60(6), 1551 – 1564.

O.B. Jennings, A. Mandelbaum, W.A. Massey and W. Whitt (1996) Serverstaffing to meet time-varying demand. Manag. Sci. 42(10), 1383 –1394.
















Motivation: Service-Rate Controls

Problem: systems with only a few servers or with inflexible staffing.

In many applications, it is possible to change the processing rate.

Example (use a service-rate control)

1 Hospital Surgery Rooms

2 Airport Security Inspection Lines























Our Contributions

We use simulation to study service-rate controls to stabilize performancein a single-server queue with time-varying arrival rates.

We conduct simulation experiments to evaluate the performance ofalternative service-rate controls.

We develop an efficient algorithm for simulating a time-varyingqueue with a service-rate control.


Our Contributions





Our Contributions





Our Contributions





The Model


The Gt/Gt/1 queue

Gt/Gt/1 Single-Server Queueing Model

Single server

Time-varying arrival rate function

First-Come First-Served service policy

Unlimited waiting space

Service rate is subject to control

i.i.d. service requirements separate from the service rate


The Gt/Gt/1 queue


Single server







The Gt/Gt/1 queue


Single server







The Gt/Gt/1 queue


Single server







The Gt/Gt/1 queue


Single server







The Gt/Gt/1 queue


Single server







The Gt/Gt/1 queue


Single server







The Arrival Process

A time-transformation of a stationary counting process:

A(t) ≡ Na(Λ(t)) ≡ Na(

∫ t

0λ(s) ds), t ≥ 0, (1)

where

Λ is the cumulative arrival rate function:Λ(t) =

∫ t0 λ(s) ds, t ≥ 0.

Na is a rate-1 counting process with unit jumps.

check: E [A(t)] = E [Na(Λ(t))] = Λ(t) =∫ t0 λ(s)ds.

All the stochastic variability is separated from the deterministic arrivalrate function.


The Arrival Process


A(t) ≡ Na(Λ(t)) ≡ Na(

∫ t

0λ(s) ds), t ≥ 0, (1)

where


∫ t0 λ(s) ds, t ≥ 0.





The Arrival Process


A(t) ≡ Na(Λ(t)) ≡ Na(

∫ t

0λ(s) ds), t ≥ 0, (1)

where


∫ t0 λ(s) ds, t ≥ 0.





The Arrival Process


A(t) ≡ Na(Λ(t)) ≡ Na(

∫ t

0λ(s) ds), t ≥ 0, (1)

where


∫ t0 λ(s) ds, t ≥ 0.





The Arrival Process


A(t) ≡ Na(Λ(t)) ≡ Na(

∫ t

0λ(s) ds), t ≥ 0, (1)

where


∫ t0 λ(s) ds, t ≥ 0.





The Service Process

Queue Length and Departure Process

Q(t) ≡ A(t)− D(t), t ≥ 0, (2)

D(t) ≡ Ns(

∫ t

0µ(s)1{Q(s)>0} ds), t ≥ 0, (3)

where

Ns is a rate-1 counting process with unit jumps, independent of Na.

E [D(t)|Q(s), 0 ≤ s ≤ t] =∫ t0 µ(s)1{Q(s)>0} ds.

The service requirement process Ns is separated from thedeterministic service-rate function µ(t).


The Service Process


Q(t) ≡ A(t)− D(t), t ≥ 0, (2)

D(t) ≡ Ns(

∫ t

0µ(s)1{Q(s)>0} ds), t ≥ 0, (3)

where


E [D(t)|Q(s), 0 ≤ s ≤ t] =∫ t0 µ(s)1{Q(s)>0} ds.



The Service Process


Q(t) ≡ A(t)− D(t), t ≥ 0, (2)

D(t) ≡ Ns(

∫ t

0µ(s)1{Q(s)>0} ds), t ≥ 0, (3)

where


E [D(t)|Q(s), 0 ≤ s ≤ t] =∫ t0 µ(s)1{Q(s)>0} ds.



The Service Process


Q(t) ≡ A(t)− D(t), t ≥ 0, (2)

D(t) ≡ Ns(

∫ t

0µ(s)1{Q(s)>0} ds), t ≥ 0, (3)

where


E [D(t)|Q(s), 0 ≤ s ≤ t] =∫ t0 µ(s)1{Q(s)>0} ds.



The Service Process


Q(t) ≡ A(t)− D(t), t ≥ 0, (2)

D(t) ≡ Ns(

∫ t

0µ(s)1{Q(s)>0} ds), t ≥ 0, (3)

where


E [D(t)|Q(s), 0 ≤ s ≤ t] =∫ t0 µ(s)1{Q(s)>0} ds.



The Service Process


Q(t) ≡ A(t)− D(t), t ≥ 0, (2)

D(t) ≡ Ns(

∫ t

0µ(s)1{Q(s)>0} ds), t ≥ 0, (3)

where


E [D(t)|Q(s), 0 ≤ s ≤ t] =∫ t0 µ(s)1{Q(s)>0} ds.



The Service-Rate Controls

Rate-matching control

µ(t) ≡ λ(t)

ρ, t ≥ 0. (4)

PSA-based square-root control

µ(t) ≡ λ(t) +λ(t)

2

(√1 +

ζ

λ(t)− 1

), t ≥ 0, (5)

based on the PSA approximation:E [W (t)] ≈ ρ(t)V /µ(t)(1− ρ(t)) = λ(t)V /(µ(t)2 − µ(t)λ(t)).

Supporting treory in Whitt (2015)




µ(t) ≡ λ(t)

ρ, t ≥ 0. (4)


µ(t) ≡ λ(t) +λ(t)

2

(√1 +

ζ

λ(t)− 1

), t ≥ 0, (5)






µ(t) ≡ λ(t)

ρ, t ≥ 0. (4)


µ(t) ≡ λ(t) +λ(t)

2

(√1 +

ζ

λ(t)− 1

), t ≥ 0, (5)






µ(t) ≡ λ(t)

ρ, t ≥ 0. (4)


µ(t) ≡ λ(t) +λ(t)

2

(√1 +

ζ

λ(t)− 1

), t ≥ 0, (5)






µ(t) ≡ λ(t)

ρ, t ≥ 0. (4)


µ(t) ≡ λ(t) +λ(t)

2

(√1 +

ζ

λ(t)− 1

), t ≥ 0, (5)




Generating the Arrival Process

Let Ak and Tk be arrival times of processes A and Na

Ak = Λ−1(Tk). (6)

Problem: need to compute Λ−1 for each arrival in each simulationrun.

Compute the inverse function Λ−1 for one cycle outside of simulationand do table lookup when simulating.

Λ−1(kC + t) = kC + Λ−1(t) for 0 ≤ t ≤ C , where C is the length of acycle.

(See the paper for the details.)




Ak = Λ−1(Tk). (6)








Ak = Λ−1(Tk). (6)








Ak = Λ−1(Tk). (6)






Generating the Service Times

Let Ak , Bk and Dk be customer’s arrival time, begin service time anddeparture time; Vk and Wk be customer’s service time and waitingtime in queue.

Let sequence of service requirements {Sk : k ≥ 1} be specified asthe times between events in the counting process Ns .

We have the basic recursions:Bk = max{Dk−1,Ak}, Dk = Bk + Vk and Wk = Bk − Ak .

But Vk is not formulated.



























Exact service time formula:

Sk =

∫ Bk+Vk

Bk

µ(s) ds, k ≥ 1. (7)

If we let

M(t) ≡∫ t

0µ(s) ds, t ≥ 0, (8)

thenVk = M−1(Sk + M(Bk))− Bk , k ≥ 1. (9)




Sk =

∫ Bk+Vk

Bk

µ(s) ds, k ≥ 1. (7)

If we let

M(t) ≡∫ t

0µ(s) ds, t ≥ 0, (8)





Sk =

∫ Bk+Vk

Bk

µ(s) ds, k ≥ 1. (7)

If we let

M(t) ≡∫ t

0µ(s) ds, t ≥ 0, (8)



Simulation Experiments


Simulation Experiments: Arrival Rates

The arrival process has the sinusoidal arrival rate function

λ(t) ≡ 1 + β sin (γt) (10)

with

β=0.2, γ=0.001, 0.01, 0.1, 1, 10.

To cover a range of difference cycle lengths of 2π/γ.




λ(t) ≡ 1 + β sin (γt) (10)

with

β=0.2, γ=0.001, 0.01, 0.1, 1, 10.





λ(t) ≡ 1 + β sin (γt) (10)

with

β=0.2, γ=0.001, 0.01, 0.1, 1, 10.





λ(t) ≡ 1 + β sin (γt) (10)

with

β=0.2, γ=0.001, 0.01, 0.1, 1, 10.



Simulation Experiments: Stochastic Components

Use renewal processes with mean 1 for the base process Na and Ns , andconsider three different i.i.d. interval time distributions.

exponential (c2 = 1)

hyperexponential (mixture of two exponentials, c2 > 1)

Erlang (sum of two i.i.d. exponentials, c2 = 0.5)




















Simulation and Estimation Methods

Consider a fixed time interval [0, T] with T= 2× 104 for γ = 0.001and T= 2× 103 for the other values of γ.

For each simulation replication, calculate performance measures atdeterministic times dt, 2dt, 3dt,...T.

Compute virtual waiting time W(t) and number of customers in systemQ(t)

Generate 10,000 independent replications to estimate mean valuesand to construct confidence intervals of performance measures.

Take the average over all replications to estimate E(W(t)) and E(Q(t))

Use t statistics to construct 95% confidence intervals for E(W(t)) andE(Q(t))


























































Simulation Results


Simulation Results: The Rate-Matching Control

1. γ=0.001

Cycle length is 6.28× 103.The left graph is Markovian model; the right graph shows (H2/H2),(H2/E2) and (E2/E2).E(Q(t)) stabilized at target, but E(W(t)) is periodic.

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

1

2

3

4

5

time in units of 104

γ = 0.001, β = 0.2, ρ = 0.8

0

arrival ratemean numbermean wait

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

2

4

6

8

10

12

14

16


γ = 0.001, β = 0.2, ρ = 0.8

0

arrival rateEQ(t) in H2H2EQ(t) in H2E2EQ(t) in E2E2



1. γ=0.001Cycle length is 6.28× 103.The left graph is Markovian model; the right graph shows (H2/H2),(H2/E2) and (E2/E2).E(Q(t)) stabilized at target, but E(W(t)) is periodic.

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

1

2

3

4

5


γ = 0.001, β = 0.2, ρ = 0.8

0


0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

2

4

6

8

10

12

14

16


γ = 0.001, β = 0.2, ρ = 0.8

0




2. γ=0.1

Cycle length is 62.8, only last 3 to 4 cycles are displayed.E(Q(t)) stabilized at target, but E(W(t)) is periodic.

1800 1850 1900 1950 2000

1

2

3

4

5

time

γ = 0.1, β = 0.2, ρ = 0.8

0


1800 1850 1900 1950 2000

2

4

6

8

10

12

14

16

time

γ = 0.1, β = 0.2, ρ = 0.8

0

arrival rateEQ(t) in H2H2

EQ(t) in H2E2EQ(t) in E2E2



2. γ=0.1

Cycle length is 62.8, only last 3 to 4 cycles are displayed.E(Q(t)) stabilized at target, but E(W(t)) is periodic.

1800 1850 1900 1950 2000

1

2

3

4

5

time

γ = 0.1, β = 0.2, ρ = 0.8

0


1800 1850 1900 1950 2000

2

4

6

8

10

12

14

16

time

γ = 0.1, β = 0.2, ρ = 0.8

0

arrival rateEQ(t) in H2H2

EQ(t) in H2E2EQ(t) in E2E2



3. γ=10

Cycle length is 0.63, only last 3 cycles are displayed.By Whitt (1984) the system converges to stationary case.

1999 2000

1

2

3

4

5

time

γ = 10, β = 0.2, ρ = 0.8

0


1999 2000

2

4

6

8

10

12

14

16

time

γ = 10, β = 0.2, ρ = 0.8

0




3. γ=10

Cycle length is 0.63, only last 3 cycles are displayed.By Whitt (1984) the system converges to stationary case.

1999 2000

1

2

3

4

5

time

γ = 10, β = 0.2, ρ = 0.8

0


1999 2000

2

4

6

8

10

12

14

16

time

γ = 10, β = 0.2, ρ = 0.8

0



Simulation Results: The Square-Root Control

1. γ=0.001

Cycles are long, and arrival rates change slowly, thus PSA isappropriate. [Whitt, 1991]E(W(t)) is stabilized, while E(Q(t)) is periodic.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

1

2

3

4

5

6


γ = 0.001, β = 0.2, ζ = 1


0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

5

10

15

20

25


γ = 0.001, β = 0.2, ζ = 1

arrival rateEW(t) in H2H2

EW(t) in H2E2EW(t) in E2E2



1. γ=0.001Cycles are long, and arrival rates change slowly, thus PSA isappropriate. [Whitt, 1991]E(W(t)) is stabilized, while E(Q(t)) is periodic.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

1

2

3

4

5

6


γ = 0.001, β = 0.2, ζ = 1


0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

5

10

15

20

25


γ = 0.001, β = 0.2, ζ = 1





2. γ=0.1

PSA does not hold as cycles are short.Neither E(W(t)) nor E(Q(t)) is stabilized.

1800 1850 1900 1950 20000

1

2

3

4

5

6

time

γ = 0.1, β = 0.2, ζ = 1


1800 1850 1900 1950 20000

5

10

15

20

25

time

γ = 0.1, β = 0.2, ζ = 1





2. γ=0.1PSA does not hold as cycles are short.Neither E(W(t)) nor E(Q(t)) is stabilized.

1800 1850 1900 1950 20000

1

2

3

4

5

6

time

γ = 0.1, β = 0.2, ζ = 1


1800 1850 1900 1950 20000

5

10

15

20

25

time

γ = 0.1, β = 0.2, ζ = 1




Thank You!



Construct the table of Λ−1 over an cycle.

Calculate Λ value for nx equally spaced points of [0,C ], let spacing beη = C/nx .

Construct approximation J of Λ−1 over ny equally spaced points in[0,Λ(C] = [0,C ], let spacing be δ = C/ny .

Using J(jδ) = kη, 1 ≤ j ≤ ny ,where 0 ≤ k ≤ nx and kη is closest point greater equal to the trueinverse value.

Extend J to [0,C ] by letting J(t) = J(bt/δcδ).






























Results From [Whitt, 2015]: The Rate Matching Control

Theorem 2.1 (time transformation of a stationary model)

For (A,D,Q) with the rate-matching service-rate control and thestationary single-server model (A1,D1,Q1),

(A(t),D(t),Q(t)) = (A1(Λ(t)),D1(Λ(t)),Q1(Λ(t))), t ≥ 0. (11)

Theorem 3.2 (stabilizing the queue-length distribution and thesteady-state delay probability)

Let Q1(t) be the queue length process when λ(t) = 1, t ≥ 0. IfQ1(t)⇒ Q1(∞) as t →∞, where P(Q1(∞) <∞) = 1, then also

Q(t)⇒ Q1(∞) in R as t →∞, (12)

andP(W (t) > 0) = P(Q(t) ≥ 1)→ ρ as t →∞. (13)








Q(t)⇒ Q1(∞) in R as t →∞, (12)

andP(W (t) > 0) = P(Q(t) ≥ 1)→ ρ as t →∞. (13)








Q(t)⇒ Q1(∞) in R as t →∞, (12)

andP(W (t) > 0) = P(Q(t) ≥ 1)→ ρ as t →∞. (13)


Results From [Whitt, 2015]: The Square-Root Control

Section 6.3 (stabilizing the expected time-varying virtual waiting time)

We assume that the Pointwise Stationary Approximation (PSA) isappropriate. Then the square-root control (14) stabilizes E [W (t)] at thetarget w for all t under heavy traffic.

µ(t) ≡ λ(t) +λ(t)

2

(√1 +

ζ

λ(t)− 1

), t ≥ 0, (14)

where ζ is inversely proportional to w .

Pointwaise Stationary Approximation (PSA)

Performance at different times can be regarded as approximately the sameas the performance of the stationary system with the model parametersoperating at those separate times.


Results From [Whitt, 2015]: The Square-Root Control

Section 6.3 (stabilizing the expected time-varying virtual waiting time)

We assume that the Pointwise Stationary Approximation (PSA) isappropriate. Then the square-root control (14) stabilizes E [W (t)] at thetarget w for all t under heavy traffic.

µ(t) ≡ λ(t) +λ(t)

2

(√1 +

ζ

λ(t)− 1

), t ≥ 0, (14)

where ζ is inversely proportional to w .

Pointwaise Stationary Approximation (PSA)

Performance at different times can be regarded as approximately the sameas the performance of the stationary system with the model parametersoperating at those separate times.


Theorem From [Whitt, 2015]

Theorem 6.1 (impossibility of stabilizing both the waiting timedistribution and the mean number in queue)

Consider a Gt/Gt/1 system starting empty in the distant past. Supposethat a service-rate control makes P(W (t) > x) independent of t for allx ≥ 0. Then the only arrival rate functions for which the mean numberwaiting in queue E [(Q(t)− 1)+] is constant, independent of t, are theconstant arrival rate functions.











Theorem 1 (Convergence of point processes)

The point process D(t) has predictable stochastic intensity Λ(t), then itcan be represented as the random-time transformation

D(t) = Π(C (t)), t ≥ 0, (15)

where Π(t) is a Poisson process with unit intensity and C (t) =∫ t0 Λ(u)du.

If Cn(t)⇒ ct in R as n→∞ for each t, then Dn ⇒ Πc in D[0,∞) asn→∞, where Πc is a Poisson process with intensity c .

Section 3.2 (Applications to queues)

Apply rescaling

Dn(t) = Dn(t)(t/n) and Cn(t) = Cn(t)(t/n) (16)

for t ≥ 0.





D(t) = Π(C (t)), t ≥ 0, (15)




Apply rescaling


for t ≥ 0.





D(t) = Π(C (t)), t ≥ 0, (15)




Apply rescaling


for t ≥ 0.





D(t) = Π(C (t)), t ≥ 0, (15)




Apply rescaling


for t ≥ 0.


References

M. Defraeye and I. Van Niewenhuyse (2013) Controlling excessive waiting times insmall service systems with time-varying demand: an extension of the ISA algorithm.Decision Support Systems 54(4), 1558 – 1567.

B. He, Y. Liu and W. Whitt (2015) Staffing a service system with non-Poissonnonstationary arrivals. Working Paper.

O.B. Jennings, A. Mandelbaum, W.A. Massey and W. Whitt (1996) Server staffingto meet time-varying demand. Manag. Sci. 42, 1383 –1394.

Y. Liu and W. Whitt (2012) Stabilizing customer abandonment in many-serverqueues with time-varying arrivals. Oper. Res. 60(6), 1551 – 1564.

W. Whitt (1984) Departures from a queue with many busy servers. Math. Oper.Res. 9(1984), 534–544.

W. Whitt (1991) The pointwise stationary approximation for Mt/Mt/s queues isasymptotically correct as the rates increase. Management Science 37(3), 307–314.

W. Whitt (2015) Stabilizing performance in a single-server queue with time-varyingarrival rate. Queueing Systems 81, 341–378.

G. Yom-Tov and A. Mandelbaum (2014) Erlang-R: A time-varying queue withreentrant customers, in support of healthcare staffing. Manufacturing and ServiceOper. Management 16(2), 283 – 299.


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USING SIMULATION TO STUDY SERVICE-RATE CONTROLS TO …ww2040/wsc_Ni_120515.pdf · 2015-12-05 · We use simulation to study service-rate controls to stabilize performance in a single-server