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誰想計算等候時間？Computing Waiting Time! Who Cares?

政治大學應用數學系陸行

Hsing Luh Department of Mathematical Sciences

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貓空纜車假日試辦「進站時間號碼牌」管制「進站時間號碼牌」管制措施是以 10 分鐘為單位，每單位設

定運送遊客 200 人。民眾只要告知工作人員欲搭乘人數，就可領到一張號碼牌，牌上載明進站時段，在該時段回到入口處就可進站搭車；逾 10 分鐘以上須重抽牌。

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多線排隊

4

單線排隊多人服務

5

等候時間的計算與應用

•單線、多線排隊

•如何描述等候時間

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如何描述等候時間

• 平均等候時間

等候時間的總和總人數

• 等候時間的變異數

• 等候時間的機率分配函數

Agner Krarup Erlang (1878-1929)

7

1−nA nA

nτ

1−nW 1−nS nS

1−nT

2−nd 1−nd nd

到達時間是隨機變數

1−nA 'nA

'nτ

1−nW 1−nS 1−nI nS

1−nT

2−nd 1−nd nd

服務時間是隨機變數

9

nnn WST +=

Pr { tWn<}

等候時間是隨機變數

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An M/M/1 Queueing systemAn M/M/1 Queueing system

P P (exact 1 arrival in [t, t+(exact 1 arrival in [t, t+∆∆t])= t])= λ∆λ∆t t P P (no arrivals in [t, t+(no arrivals in [t, t+∆∆t])= 1-t])= 1-λ∆λ∆tt P P (1 service completion in [t, t+(1 service completion in [t, t+∆∆t])= t])= µ∆µ∆t t P P (no service completion in [t, t+(no service completion in [t, t+∆∆t])= 1-t])= 1-

µ∆µ∆tt PPnn(t) = (t) = P P (# of (# of customerscustomers is n at time t) is n at time t)

Courtesy to Internet Information

11

)()()()()(

)()()()()()()(

0,110,000

,11,11,

tatPtatPttP

tatPtatPtatPttP nnnnnnnnnn

∆+∆=∆+∆+∆+∆=∆+ ++−−

ttPttPttP

ttPttPtttPttP nnnn

∆+∆−=∆+∆+∆+∆−∆−=∆+ +−

µλµλµλ

)()1)(()(

)()()1)(1)(()(

100

11

)()()(

),()()()()(

100

11

tPtPdt

tdP

tPtPtPdt

tdPnnn

n

µλ

µλµλ

+−=

+++−=+−

狀態平衡方程式


12

n-1 n n+1

λ λ λ λ

µ µ µ µ

)()()(

),()()()()(

100

11

tPtPdt

tdP

tPtPtPdt

tdPnnn

n

µλ

µλµλ

+−=

+++−=+−

1,0)0(

0.1)0(0

≥=

=

nP

P

n Courtesy to Internet Information

狀態平衡方程式

13

長時間的觀察值長時間的觀察值

1 1

0 1

0 ( ) ,

0

n n nP P P

P P

λ µ λ µ

λ µ

− += − + + +

= − +

n-1 n n+1

λ λ λ λ

µ µ µ µ

,lim ( )

n i ntP P t

→∞=


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基本假設條件基本假設條件

Kirchov’s current conservation lawKirchov’s current conservation lawFlow in = Flow outFlow in = Flow out

PP00++PP11+…++…+PPNN=1=1


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區域狀態平衡方程式區域狀態平衡方程式

λλPP0 0 ==µµ PP11



……....

λλPPn-1 n-1 ==µµ PPnn

0 1 2 3 n-1 n n+1

λ λ λ λ λ λ λ λ

µ µ µ µ µ µ µ µ


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機率分配機率分配

PP11= ( = ( λλ//µµ))PP00

PP22= (= (λλ//µµ))PP11

……....

PPnn= (= (λλ//µµ))PPn-1n-1

∑∞

=

=

=

0

0

0

)(

1

)(

n

n

nn

P

PP

µλ

µλ


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Little’s LawLittle’s Law

The mean number of customer in a The mean number of customer in a queueing system queueing system L=L=λλWW W is the mean waiting timeW is the mean waiting time


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Example of Little LawExample of Little Law

The average waiting time for M/M/1 queueThe average waiting time for M/M/1 queue

Little Law also holds if we consider only the Little Law also holds if we consider only the queue and not the server.queue and not the server.

)1(

/1

ρµ

−=W

)1(/1

)1(

/1

ρλµ

ρµ

−=−

−=qW


19

State dependent M/M/1 Queueing Queueing SystemsSystems

0 1 2 3 n-1 n n+1

λ(0) λ(1) λ(2) λ(3) λ(n-2) λ(n-1) λ(n) λ(n+1)

µ(1) µ(2) µ(3) µ(4) µ(n-1) µ(n) µ(n+1) µ(n+2)

∑∞

= =

−∏+=

1 1

0

)()1(

1

1

n

n

i ii

P

µλ


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Performance MeasurePerformance Measure Mean Throughput of the queueing systemMean Throughput of the queueing system

Mean rate at which customers pass throughMean rate at which customers pass through

Mean number of customers in the queueing Mean number of customers in the queueing systemsystem

Mean time delay (using Little’s Law)Mean time delay (using Little’s Law)

UtilizationUtilizationU=1-pU=1-p00

∑∞

=

=1

)(n

nPnY µ

∑∞

=

=1n

nnPL

∑

∑∞

=

∞

===

1

1

)(n

n

nn

Pn

nP

Y

LW

µ


21

Example 1

1/ 1

(1 )

(1 )

q

q

W W

W

µρ µ

λρ

= = +−

=−

hrjobsjobs /5min/12

1 ==λ hrjobsjobs /6min/10

1 ==µ

Suppose interarrival time is 12 min and service time is 10 minutes per job.

.

M/M/1:

0 1 ( / )p λ µ= −

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M/M/1/N Queueing SystemM/M/1/N Queueing SystemThe Finite Buffer CaseThe Finite Buffer Case

Blocking probabilityBlocking probability

λ

0 1 2 3 N-1 N

λ λ λ λ λ

µ µ µ µ µ µ

n

Nn

NN

n

n

nn

p

p

Nnpp

)()(1

1

)(1

1

)(

1

0,)(

1

1

0

0

0

µλ

µλ

µλ

µλ

µλ

µλ

µλ

+

+

=

−

−=∴

−

−==

≤≤=

∑


23

Example 2

2

2

2

1qWρρ

=−

hrjobsjobs /5min/12


1 ==µ


.M/M/2:

0

1

1

2

pρρ

λρµ

−=+

=

24

比比看

Two M/M/1

One M/M/2

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平均等候時間

平均服務率 µ2

λM/M/1

平均到達率

P{idle time}

E(waiting time) ( )qW2

λµµ λ

=−

µλµ

2

2P0

−=

× 2

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平均等候時間

平均服務率 µ λM/M/2

平均到達率

P{idle time}}

E(waiting time}

λµλµ

+−=

2

2p0

( ) ( )λµλµµλ

−+=

22W

2

q

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Example 3

hrjobsjobs /5min/12


1 ==µ


.

Two M/M/1:

One M/M/2:

× 2

1120 min

job1 1

5 12

q

10.0714( )

(2 ) 14

1L 0.003 (jobs)

2 2 336

qWλµµ λ

λ λµµ λ

= = = ≈− −

= = ≈−

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M/M/2 與 M/M/1 的比較

M/M/2 v.s M/M/1

E(wainting_time) ( ) ( )λµλµµλ

−+ 22

2

( )λµµλ

−2

P{idle_time} λµλµ

+−

2

2

µλµ

2

2 −

Var(waiting_time) ( )

( ) ( ) 222

222

22

24

λµλµµλλµµλ

−+−+

( )( ) 22 2

4

λµµλµλ

−−

Var(system_time) ( ) ( ) 22

323

22

2416

λµλµµλµλµ

−++−

( ) 22

4

λµ −

<

<

<

× 2

× 2

in queue) <

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等候時間的機率分配函數M/M/2

≥≤≤

=2n,2

2n,1nn µ

µµ , λ ,

µλγ =

µλγρ

2c==

30

0 0.5 1 1.5 2 2.5 3 3.5 40

10

20

30

40

50

60

y = exp(x)

y = x

y

x

31

佇列的等候機率 M/M/2 v.s M/M/1

P{waiting time > t min} ( )( )λµ

λµµλ -2t-

2

e2 +

2-t-

e2

λµ

µλ

32

時間延誤問題

系統模擬

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系統模擬 (1)

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系統模擬 (2)

35

系統模擬 (3)

36

交通號誌

37

匝道管制

38

通訊問題

39

通訊服務

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An optimization model

Minimize Penalty made by unsatisfied traffic aggregates

Subject toFixed resource (Limited Short Path bandwidth)Traffic aggregate QDF budget must be satisfiedAllocate resource to traffic aggregates

Trading off QDF with bandwidth in each Short Path

最佳化模型

41

Introduction

They tried to get the optimal solution of the model as follows:

..

42

Utility Functions

An utility function of (the bandwidth of each class ):

where aspiration and reservation levels,

and , respectively, and .

iq

i

iii r

qqf

iδlog)( =

iii ra=δiria

,

43

Utility Functions

The graph of an utility function : )( ii qf

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A Network Optimization Model

45

A Network Optimization Model

is a set satisfying constraints of the Model.*Q

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Network Constraints

To determine whether link e is chosen we define the binary decision variable:

For each connection j of class i, we denote the

routing path connecting the source node o and destination node d by .

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Network Constraints

It shows that every connection in the same class uses the same bandwidth and has the same bandwidth requirement.

Iibq ii ∈∀≥ ,

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Network Constraints

We have the budget constraint on the network:

Since every connection has the maximal end-to-end delay constraint.

∑ ∑ ∑∈ ∈ ∈

≤⋅⋅Ee Ii Kj

jiiei

Beq )(,χκ

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Network Constraints

Flow constraints:

50

An Illustrative Example

Network Topology for an Illustrative Example

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:

52


QoS: Quality of Service.

53

Numerical Results

54

Numerical Results

55

Numerical Results

56

Numerical Results

57

Numerical Results

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參考文獻

Hsing Luh, Chia-Hung Wang, and Chih-Chin Liang, “Measuring the Cost of Links of a Disaster-Avoided Content-Delivering Service within a Large-scaled Company,” Proc. of the Fourth International Conference on Queueing Theory and Network Applications (QTNA2009), Fusionopolis, Singapore, July 29-31, 2009.

Hsing Luh and Chia-Hung Wang, “A Fast Branch and Bound Based Algorithm for Bandwidth Allocation and QoS Routing on Class-based IP Networks,” Proc. of the First World Congress on Global Optimization in Engineering & Science (WCGO2009), Hunan, China, June 1-5, 2009.

Hsing Luh and Chia-Hung Wang, “Measuring Average Revenue with Bandwidth and Traffic Demand on Communication Networks,” Proc. of International Conference on Computational Intelligence and Software Engineering (CiSE 2009), Wuhan, China, December 11-13, 2009.

Hsing Luh, Chih-Chin Liang, Chia-Hung Wang, Ping-Yu Hsu, and Wuyi Yue, “Proactive Peer-to-Peer Traffic Control when Delivering Large Amounts of Content within a Large-Scale Organization,” Proc. of the Ninth International Conference on Next Generation Wired/Wireless Networking (NEW2AN 2009), St. Petersburg, Russia, September 15-18, 2009.

Ji-hong Li, Nai-shuo Tian, Zhe George Zhang, Hsing Paul Luh, “Analysis of the M/G/1 queue with exponentially working vacations —A matrix analytic approach.” Queueing Systems, Vol. 61, 2009, 139-166. DOI 10.1007/s11134-008-9103-8.

Chia-Hung Wang and Hsing Luh, “A Utility Maximization Scheme for End-to-End QoS Routing in Communication Networks with Heterogeneous Service Classes,” 2008 IEEE International Conference on Service Operations and Logistics, and Informatics, Beijing, Oct, 2008.

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作業研究

所謂「運籌」，就是「調度」，就是將有限資源做有效及最佳調度與配置的意思。

RRf n →:)(x

資源最佳值

有效調度配置

目標

例如：一個企業管理者必須將原料、半成品、成品、存貨、勞動力、資金、時間，做最佳的調度與組合，期使成本降到最低，利潤增到最

大。

60

運籌科學與相關領域 OR and Related Fields

•作業研究作業研究

•運籌學運籌學

•Operations ResearchOperations Research

•數學系、應用數學系數學系、應用數學系 Department of Mathematics, Department Department of Mathematics, Department of Mathematical Sciencesof Mathematical Sciences

•工業工程系、資訊科學系工業工程系、資訊科學系Department of Industrial Engineering, Department of Industrial Engineering, Department of Computer ScienceDepartment of Computer Science

•工商管理系、統計系工商管理系、統計系 Department of Business Administration, Department of Statistics

•線性規劃、非線性規劃、動態規劃線性規劃、非線性規劃、動態規劃 Linear Programming, Nonlinear Programming, Dynamic Linear Programming, Nonlinear Programming, Dynamic ProgrammingProgramming

•等候理論、隨機規劃、系統模擬等候理論、隨機規劃、系統模擬 Queueing Theory, Stochastic Programming, System Simulation

61

運 OR 籌資訊

Information

管理管理 ManagementManagement

科學 Science

62 By Bertrand Mareschal

1. Introduction

History.

Methodology. Models.

2. Linear programming

Applications.

Simplex algorithm.

Sensitivity analysis.

Integer programming.

3. Network analysis

Transportation problems.

Maximum flow problems.

CPM and PERT – Project management.

4. Inventory management

Deterministic EOQ models.

Probabilistic models.

Production management

5. Multicriteria decision aid

Methods.

Multicriteria decision support systems.

Group decision systems.

Applications.

63

模式與方法 Models and Methodologies

Models deterministic probabilistic -------------------------------------------------- optimization / OR OR solutions \ Methodologies simulation --------------------------------------------------

64 By Bertrand Mareschal

Some of the most widely used techniques : Linear programming :

Optimization under constraints

Production planning, scheduling, financial planning,

Simulation methods :

Simulate complex systems

Queueing

PERT/CPM :

Complex projects management

Network models :

Transportation problems

Inventory management :

Reduce inventory levels and costs

JIT production management

65

Statistical analysis :

Summarize and analyze data.

Multicriteria decision aid :

Identify best compromise solutions

When conflicting objectives are considered

Group decision aid :

Achieve consensus efficiently

Dynamic programming :

Repetitive (w.r.t. time) decision problems.

Reliability theory :

Reliability of equipments.

66

供應鏈 Supply Chain Management

The Goal: A Process of Ongoing Improvement

by

Eliyahu M. Goldratt

67

推銷員問題 Traveling Salesman Problem TSP

http://www.ing.unlp.edu.ar/cetad/mos/TSPBIB_home.html

O(2^n)

2^50 ~10^15

3*10^7*10^610^6~ 30 years

68

BIOINFORMATICS 生物資訊

http:// motif.stanford.edu/index.html

69

CRYTOGRAPHY 密碼學

Elliptic Curve http:// www.certicom.ca

70

( ) : n ntf R R→x

資源 Resource

最佳值 Optimum

有效調度 Adjustment

配置 Allocation

目標Objective

動態系統

71

最佳解

改進

最佳解

可行解

72

No interac-tion allowed

2

1

Individual sampling model (ISM): Simulated patient-level Markov model (SPLMM) (variations as in quadrant below for patient level models with interaction)

Simulated Markov model (SMM) Markov model (evaluated deterministically)

Timed

Individual sampling model (ISM): Simulated patient-level decision tree (SPLDT)

Simulated decision tree (SDT)Decision tree rollback

Untimed

Non-Markovian, discrete-state,

individuals

Markovian, discrete state,

individuals

Markovian, discrete state, stochastic

Expected value, continuous

state, deterministic

Individual level Cohort/aggregate level/counts

DCBA

73

Discrete event simulation (CT, DES)

Discrete individual simulation (DT, DES)

Interaction allowed

4

3

Continuous time Individual event history model (CT, IEH)

Continuous time Markov chain model (CTMC)

System dynamics (ordinary differential equations, ODE)

Continu-ous time

Discrete-timeindividual eventhistory model(DT, IEH)

Discrete time Markov chain model (DTMC)

System dynamics (finite difference equations, FDE)

Discrete time

Non-Markovian, discrete-state,

individuals

Markovian, discrete state,

individuals

Markovian, discrete state, stochastic

Expected value, continuous state,

deterministic

Individual level Cohort/aggregate level/counts

DCBA

74

Queueing Theory -- Establishing policies for sending in resupply aircraft to small airfields with maximum on ground (MOG) limits. For

example, the airfield at Haiti had a MOG of 3 C-5s.

Decision Analysis -- A recent study was completed that looked at

methods for considering the counter proliferation effects of weapons

of mass destruction when evaluating new systems.

Response Surface Methodology -- Using multiple runs of a large

theater-level model, Thunder, to build a response surface allowing

quick analysis of effects of changes in air apportionment.

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謝謝大家

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