Strategic Timing Decisions in Service Systems · Strategic Timing Decisions in Service Systems. Games of Timing Individual timing decisions in service systems: When should I join

Nahum ShimkinDepartment of Electrical Engineering

Technion

Workshop on Congestion Games IMS/NUS, Singapore, December 2015

Strategic Timing Decisionsin Service Systems

Games of Timing

Individual timing decisions in service systems:

When should I join a queue?

When do I abandon a queue?

When should I start a ride to work?

When our decision affects the utility of others (and vice-versa), we obtain a timing game.

2Nahum Shimkin, Technion

Nahum Shimkin, Technion3

Abandonment Decisions

Rational Abandonments from Invisible Queues [with Avishai Mandelbaum: 2000, 2004]

Consider the standard M/M/m+G queue:

is the patience distribution

How is 𝐺 ⋅ formed? How is it related to the system performance and the customer expectations?


( servers)m

~ ( )T G

( ) { }G t P T t

Empirical Observations

Hypothesis: Customers adapt their patience to the expected wait in the queue.


Average abandonment time vs. average wait (at different times of day)

Data obtained from a banking call center (a tele-queue).

0

100

200

300

400

500

600

700

800

900

1000

0 50 100 150 200 250 300 350 400 450 500

E [

Pa

tie

nc

e ]

, s

ec

E [ Offered wait | Wait>0 ], sec

Rational Abandonment Model

For each customer of type

Waiting cost (linear / convex)

Service utility

Utility function: is the offered waiting time

Abandonment time:


( )C w

R

{ }( ) ( )W W TU T E R I C W T

W

P

( servers)m

?T

0argmin ( )TT U T

Strategic Equilibrium

Nash equilibrium (NEP):

Main results:

Existence and uniqueness of the NEP

Computation of the resulting patience profile

"Inverse modeling": deduction of customer utilities from data, for counterfactual analysis


/ /ˆ ,

{ } ( ) { }W

W

M M m G F F UP

T G F T


Arrival Time Decisions

Strategic Choice of Arrival Times in Transient Queues

With Sandeep Juneja and Rahul Jain: 2011,2013

The model:

A FCFS service system that starts serving at time 0

A finite population (possibly of random size) of customers

Each customer may choose his or her arrival time to the queue


Illustration:

Lining up …

for tickets: e.g. for a football match or rock concert (early buyers get better seats)

for a new product (new iPad, Harry Potter book)

for lunch in a busy cafeteria


Decision Model

A linear cost function for each customer j :

𝑤𝑗 is j's waiting time in the queue

𝜏𝑗 his service completion time

Each customer wishes to minimize his own expected cost 𝐸(𝐶𝑗)by appropriately choosing his arrival time distribution 𝐹𝑗(𝑡)

The single-class stochastic model was first by Glazer & Hasin (1983), for 𝛽 = 0,


j j j j jC w

Homogeneous Population:

There exists a unique equilibrium distribution, , which is characterized by an FDE with boundary conditions.

General form:

In the fluid limit ( 𝑁 → ∞), 𝐹(𝑡) tends to a uniform distribution.

PoA>2, and equals 2 for the fluid model.


,j j

( ) ( )jF t F t

00

'( )F t

at bt

Heterogeneous (Multiclass) Customer Population

Fluid Model


t

'( )F t

3T

1T 2T4T

3class

1m

( )Q t2m

3m

3

3m

1classi

i iim


Timing of Ad Posting

The Timeline Game

with Eitan Altman (2015

System Model:

An ordered list ("timeline") of on-line items(ads, posts, photos, links).

Items are submitted by individual users,and initially placed at the top position.

Older items roll down when newer onesarrive.

Examples:

On-line classified ads (Craigslist, Yad2)

Discussion forums, talkbacks

Facebook posts


position 1

position 2

position K

IN

OUT

Utility Model

Utility of a posted item:

is the time spent in position k

is a time-dependent exposure function

is the relative utility parameter for position k, with


11 [ , )

( )

k k

K

k

k t T T

U r u t dt

1[ , )k kT T

( ) 0u t

kr

1 2 1 0K Kr r r r

The Game Model

The system operates over a finite time interval 0, 𝑇 .

The number of users (players) is 𝑁0 –possibly random, with distribution 𝑝0 𝑛 , 𝑛 ≥ 0.

Each player 𝑖 is free to choose the submission time 𝑡𝑖 of his/her item,with the goal of maximizing its exposure.

We consider the problem as a non-cooperative game, and seek a symmetric NEP for this game:


( ( ), [0, ])it F F t t T

0 T

( )u t

?it

Characterization

An equilibrium profile 𝐹 is characterized by the following properties:

1. The density 𝑓 𝑡 = 𝐹′(𝑡) exists, and is supportedon an interval [0, 𝐿] with 𝐿 < 𝑇.

2. 𝐹 satisfies the anti-causal FDE

where depends on 𝐹 through .


( )'( ) , [0, )

( , )

u tF t t L

g t F

( ; )g t F ( ( ), [ , ])F s s t T

0 T0

1

( )F t

t

Explicit Solutions

In case of

Poisson demand:

and exponential weights:

we obtain

PoA:


0 Pois( )N 1

1 , 1 (0 1)kkr r k

'( ) ( ) on [0, ]F t u t t L

(1 )1(1 )

1( 1 as )

(1 )(1 )Relative Efficiency

e


When to Start a Trip:The Morning Commute Problem

The Bottleneck Model

Introduced by Vickrey (1969) to model the individual timing decisions of commuters who need to get to their destination at a certain time, during rush hours.

Also known as “the morning commute problem”.

Congestion is modeled as a FCFS fluid queue – assuming a single “bottleneck”.

Many generalizations in the transportation literature – we focus here the single queue, multiclass version.


The model

Individual commuters (players) are infinitesimal, with total mass Λ>0

Each user chooses a start time (𝑡𝑠), at which he joins the queue.

The cumulative arrival process is denoted 𝐴 𝑡 = −∞𝑡𝜆 𝑠 𝑑𝑠,

with 𝐴 −∞ = 0, 𝐴 ∞ = Λ.

The queue is single-server FCFS, with constant service rate 𝜇

Hence, a player who arrives at the queue at time 𝑡𝑠 and sees a queue size𝑄 = 𝑄 𝑡𝑠 , will depart (reach the destination) at time

𝑡𝑑 = 𝑡𝑠 + 𝑄/𝜇


( )t

( )Q t

Typical Queue Buildup:


t

( )Q t

t

( )t

Cost functions

We assume a finite number 𝐾 of player classes, with class 𝑘 having mass Λ𝑘(and 𝑘 Λ𝑘 = Λ ).

The cost function of a class 𝑘 player is give by

𝐿𝑘(⋅) typically expresses a cost for being late or early relative to an ideal time. However, more general are allowed.

The following properties are assumed throughout:

1. 𝐿𝑘(⋅) is a continuous, non-negative function, and lim|𝑡|→∞𝐿𝑘 𝑡 = ∞ .

2. 𝛼𝑘 > 0, and 𝛼𝑘 + 𝐿𝑘′ (t)>0.


(linear) waiting cost + latenss cost

( , ) ( ) ( )k s d k d s k dC t t t t L t

Equilibrium

A (Nash-Wardrop) equilibrium is defined, as usual, by the requirement that no player has an incentive to deviate. More concretely:

A arrival profile is a set of arrival densities {𝜆𝑘 𝑡 }𝑘∈𝐾, with −∞∞𝜆𝑘 𝑠 𝑑𝑠 = Λ𝑘 .

Given an arrival profile, each potential arrival at a time 𝑡𝑠 has a unique departure time 𝑡𝑑 𝑡𝑠 .

An arrival profile is a NEP if (essentially), for every class 𝑘,

In particular, the cost for all players of a given class is the same.


*

( , )( ) 0 ( , ( )) min ( , ( ))

s

k k d k s d s kt

t C t t t C t t t c

Departure-time characterization

The NEP is naturally defined in terms of the decision variable 𝑡𝑠.

However, a more convenient representation is obtain by working with the departure times 𝑡𝑑. We follow the elegant formulation in Lindsey (2004).


Iso-Cost Functions

Recall our cost functions: 𝐶𝑘 𝑡𝑠, 𝑡𝑑 = 𝛼𝑘 𝑡𝑑 − 𝑡𝑠 + 𝐿𝑘 𝑡𝑑 ,which can be written in the form

𝐶𝑘 𝑤, 𝑡𝑑 = 𝛼𝑘𝑤 + 𝐿𝑘 𝑡𝑑

For each class 𝑘 and cost parameter 𝑐𝑘 ∈ −∞,∞ , define the iso-cost function 𝑊𝑘 ⋅; 𝑐𝑘 via:

These functions describe the waiting time of a class-k customer that departs the queue at 𝑡𝑑 if his total cost equals 𝑐𝑘.


( )( ; ) k k d

k d kk

c L tW t c

dt

( )k dL t

*tdt

( , )k dW t c

*t

c

Single-class Equilibrium

Suppose 𝐾 = 1.

In equilibrium, all players in this class must have the same cost, say 𝑐1∗.

𝑐1∗ is determined by: 𝑚 𝑡:𝑊1 𝑡, 𝑐 ≥ 0 = Λ1/𝜇 .

28

dt

1( , )dW t cc

1 /

dt

1( , )dW t c*1c c

Multi-Class Equilibrium

Suppose 𝐾 > 1. In equilibrium, all players of class 𝑘 must have the same cost, 𝑐𝑘∗ .

Equilibrium condition: 𝒄 = 𝑐1, … , 𝑐𝐾 satisfy, for every 𝑘 = 1,… , 𝐾:

𝑚(𝑇𝑘 𝒄 ) = Λ𝑘/𝜇,

where 𝑇𝑘 𝒄 ≜ {𝑡 ∶ 𝑊𝑘 𝑡, 𝑐𝑘 ≥ 0, 𝑊𝑘 𝑡, 𝑐𝑘 = max𝑗=1,…,𝐾

𝑊𝑗(𝑡, 𝑐𝑗)}

29

2 /A B

dt

1 1( , )dW t c 2 2( , )dW t c 3 3( , )dW t c

A B

Multi-Class Equilibrium

Non-overlap Assumption: Assume (for now) that for any constants 𝑐1, … , 𝑐𝐾 and 𝑘 ≠ 𝑗, the following sets have measure zero:

{𝑡: 𝐿𝑘 𝑡, 𝑐𝑘 = 0}, {𝑡: 𝐿𝑘 𝑡, 𝑐𝑘 = 𝐿𝑗(𝑡, 𝑐𝑗)}

Proposition: An arrival profile 𝜆𝑘 𝑡 is a NEP if and only if there exists cost parameters 𝒄 = 𝑐1, … , 𝑐𝐾 that satisfy the above equilibrium condition, namely

𝑚 𝑇𝑘 𝒄 = Λ𝑘/𝜇, 𝑘 = 1,… , 𝐾 (#)


Uniqueness

Proposition: The set of cost parameters that satisfy the equilibrium condition (#) is unique

Proof:

Consider two sets of cost parameters, 𝑐1, … , 𝑐𝐾 and 𝛾1, … , 𝛾𝐾 , that satisfy (#). Define

𝐾+ = 𝑘: 𝑐𝑘 > 𝛾𝑘 , 𝐾0 = 𝑘: 𝑐𝑘 = 𝛾𝑘 , 𝐾− = {𝑘: 𝑐𝑘 < 𝛾𝑘}.

If 𝐾+ is non-empty, then 𝑘∈K+𝑚 𝑇𝑘(𝒄) > 𝑘∈K+𝑚 𝑇𝑘(𝜸)


Computation

Computation of the NEP boils down to finding the unique cost parameters 𝑐1, … , 𝑐𝐾 that satisfy (#).

We may consider each 𝑐𝑘 as an “action” of class 𝑘, and consider this computation as finding an equilibrium point in an auxiliary game.

The best response for class 𝑘 in this game is defined as

𝐵𝑅𝑘 𝒄−𝑘 = {𝑐𝑘: 𝑚 𝑇𝑘 𝑐𝑘; 𝒄

−𝑘 =Λ𝑘

𝜇} ,

which is a singleton.

An equilibrium is a fixed point of this map.

Note: We may trivially associate this best response map with game with cost function

𝐽𝑘(𝒄) = |𝑚 𝑇𝑘 𝒄 −Λ𝑘

𝜇|. However, this is not needed.


Monotonicity

It is easily verified that the map 𝑐𝑘= 𝐵𝑅𝑘 𝒄−𝑘 is monotone non-decreasing

in each 𝑐𝑗 , 𝑗 ≠ 𝑘.

The auxiliary game therefore satisfies the essential properties of submodular games (Topkis 1979, 1998), and convergence results for such games apply.

In particular, we obtain convergence results for the standard

• Sequential best-response algorithm

• Simultaneous best-response algorithm


Sequential BR Algorithm

The algorithm:

- Initialization: Start with cost parameters 𝒄 = 𝑐1, … , 𝑐𝐾 such that 𝑚 𝑇𝑘(𝒄) ≤ Λ𝑘/𝜇, ∀𝑘.

- Repeat: Choose some 𝑘, and update 𝑐𝑘= 𝐵𝑅𝑘 𝒄−𝑘 .

Proposition: Suppose that each class 𝑘 is updated infinitely often. Then each 𝑐𝑘 converges monotonically to 𝑐𝑘

∗, the equilibrium cost parameter of the (original) game.


Sequential BR Algorithm

Proof (direct):

Initially, 𝑚 𝑇𝑘(𝒄) ≤ Λ𝑘/𝜇, ∀𝑘.

Suppose, by induction, that this is maintained throughout, and each 𝑐𝑘 is non-decreasing up to the present stage.

Let 𝑐𝑗 be updated next. Then

─ 𝑚 𝑇𝑗(𝒄) = Λ𝑗/𝜇, by definition of BR,

─ Since all other 𝑐𝑘’s have (weakly) increased since last update (or initialization) of 𝑐𝑗, then 𝑐𝑗 increases.

─ Consequently, 𝑚 𝑇𝑘(𝒄) decreases ∀𝑘 ≠ 𝑗, and 𝑚 𝑇𝑘(𝒄) ≤ Λ𝑘/𝜇 is maintained.

It follows that each 𝑐𝑘 is monotone increasing, and since it is bounded it converges to some 𝑐𝑘

0. By continuity, 𝑚 𝑇𝑘(𝒄) converges to 𝑚 𝑇𝑘(𝒄0) .

But if 𝑐𝑘 is updated infinitely often then 𝑚 𝑇𝑘(𝒄) = Λ𝑘/𝜇 infinitely often, hence 𝑚 𝑇𝑘(𝒄

0) = Λ𝑘/𝜇.


Relaxing the Non-overlap Assumption

Non-overlap Assumption: For any constants 𝑐1, … , 𝑐𝐾 and 𝑘 ≠ 𝑗, the following sets have measure zero:

{𝑡: 𝐿𝑘 𝑡, 𝑐𝑘 = 0}, {𝑡: 𝐿𝑘 𝑡, 𝑐𝑘 = 𝐿𝑗(𝑡, 𝑐𝑗)}

Without that property, it is still true that the set of equilibrium cost parameters 𝑐1∗, … , 𝑐𝐾

∗ is unique (although the equilibrium arrival profile 𝜆𝑘 𝑡 need not be so).

However, since 𝑚(𝑇𝑘 𝒄 ) need not be continuous now, the sequential BR algorithm might run into all sorts of problems…



1 21; 4, 1

40 3

Modified BR Algorithm

- Initialization: Cost parameters 𝒄 = 𝑐1, … , 𝑐𝐾 such that 𝑚 𝑇𝑘(𝒄) ≤ Λ𝑘/𝜇, ∀𝑘.

- Repeat:

- Choose some 𝑘, and update

𝑐𝑘= 𝐵𝑅𝑘 𝒄−𝑘 ≔ min{𝑐𝑘: 𝑚 𝑇𝑘 𝑐𝑘; 𝒄

−𝑘 ≥Λ𝑘

𝜇}

- If 𝑚 𝑇𝑘 𝑐𝑘−; 𝒄−𝑘 <

Λ𝑘

𝜇, implying that 𝑊𝑘 ⋅, 𝒄 coincides with some

𝑊𝑗 ⋅, 𝒄 :

- Increase 𝑐𝑘 and 𝑐𝑗 together till they exceed their joint demand. That is:

- 𝑐𝑘 ≔ 𝑐𝑘 + 𝑏, 𝑐𝑗 ≔ 𝑐𝑗 + 𝑏, where 𝑏 is the minimal number such that

- 𝑚 𝑇𝑘 𝒄 ∪ 𝑇𝑗 𝒄 ≥Λ𝑘

𝜇+Λ𝑗

𝜇

- Repeat with additional class indices if needed.


Additional Issues

Stopping criteria

Convergence rates

(Network Problems)


Nahum Shimkin, Technion 40

Best Joint Response

Consider a constrained n-person game Γ, with strategies 𝑠𝑖 ∈ 𝑆𝑖, utility functions 𝑈𝑖 𝒔 , and constraint 𝑠1, … , 𝑠𝑛 ∈ 𝑆.

A (constrained) best-response for player 𝑖 is defined as usual by

𝐵𝑅𝑖 𝑠−𝑖 = max

𝑠𝑖: 𝑠𝑖,𝑠−𝑖 ∈𝑆𝑈𝑖(𝑠𝑖 , 𝑠

−𝑖)

A best joint response to 𝒔 = 𝑠1, … , 𝑠𝑛 ∈ 𝑆 is defined as a the strategy profile 𝜎1, … , 𝜎𝑛 ∈ 𝑆 such that

𝐵𝐽𝑅 𝒔 = argmax𝝈∈𝑆(𝒔)

𝑖=1

𝑛

𝑈𝑖(𝜎𝑖 , 𝑠−𝑖}

where

𝑆 𝒔 = {𝝈 ∈ 𝑆 ∶ 𝜎𝑖 , 𝑠−𝑖 ∈ 𝑆}.


Simultaneous (Sequential) BR Algorithm

The algorithm:

- Initialization: Start with cost parameters 𝒄 = 𝑐1, … , 𝑐𝐾 such that 𝑚 𝑇𝑘(𝒄) ≤ Λ𝑘/𝜇, ∀𝑘.

- Repeat: Choose some 𝑘, and update 𝑐𝑘= 𝐵𝑅𝑘 𝒄−𝑘 .

Proposition: Suppose that each class 𝑘 is updated infinitely often. Then each 𝑐𝑘 converges monotonically to 𝑐𝑘

∗, the equilibrium cost parameter of the (original) game.


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Strategic Timing Decisions in Service Systems · Strategic Timing Decisions in Service Systems. Games of Timing Individual timing decisions in service systems: When should I join