The Concert/Cafeteria Queueing Problem A Game of Arrivals...Joint work with Rahul Jain (USC, LA) and Nahum Shimkin (Technion, Israel) ()Game of Arrivals Cambridge, June 16 3 / 30.

The Concert/Cafeteria Queueing ProblemA Game of Arrivals

Sandeep Juneja, TIFR.

Joint work with Rahul Jain (USC, LA)and Nahum Shimkin (Technion, Israel)

Cambridge, June 16

Sandeep Juneja, TIFR . Joint work with Rahul Jain (USC, LA) and Nahum Shimkin (Technion, Israel) ()Game of Arrivals Cambridge, June 16 1 / 30

Concert or Cafeteria Queueing Problem: Framework

Finite but large number of customers

Server becomes active at time zero

Customers can come and queue up before or after time zero

Customer cost is additive and linear in waiting time and servicecompletion time

Multi-class allowed in that the cost coefficients may differ.






























Concert or Cafeteria Queueing Problem: Contributions

We analyze the associated fluid model. Each customer isinfinitesimal, has negligible effect on others. This provides greatdeal of analytical tractability

We show that the game has a unique Nash equilibrium point, andexplicitly identify this point.

We show that price of anarchy equals 2 in the single classsetting. We develop tight bounds on it in multi-class settings.

We study some methods to reduce anarchy: Service timerestrictions, assigning differing priorities, charging tariffs as afunction of time of service




















Some Motivations

The hard to beat long queue at TIFR canteen

Concert, movie theater, passport and DMV office queueing: Costto going late, going early may involve large wait in queues

We also study a modification: Cost may not be a function oftime, but of number of customers that have arrived earlier


Some Motivations





Some Motivations





Long Queues


Related Literature: Strategic Timing Decisions

Extensive literature studying admissions control, routing,reneging, pricing etc. Summarized in monograph by Hassin andHaviv (2003).

Equlibrium arrival patterns with finite service periods consideredby Glazer and Hassin 1983. Number of arrivals Poisson,Exponential service times, only waiting costs

Transportation science literature: Substantial on equilibrium fluidmodels for traffic including Lindsay (2004), Newell (1987), Smith(1984), Hendrickson and Kocur (1981), Daganzo (1985, 98)


Related Literature: Strategic Timing Decisions

Extensive literature studying admissions control, routing,reneging, pricing etc. Summarized in monograph by Hassin andHaviv (2003).

Equlibrium arrival patterns with finite service periods consideredby Glazer and Hassin 1983. Number of arrivals Poisson,Exponential service times, only waiting costs

Transportation science literature: Substantial on equilibrium fluidmodels for traffic including Lindsay (2004), Newell (1987), Smith(1984), Hendrickson and Kocur (1981), Daganzo (1985, 98)


Mathematical Framework: Asymptotic Analysis

Series of queueing systems indexed by n

For system n, there are n customers. Customer i picks arrival timefrom Fi (·).

There exists an arrival profile F (t) such that

1

n

n∑i=1

Fi (nt)→ F (t)

so fraction of arrivals by time nt are stabilizing to F(t).






1

n

n∑i=1

Fi (nt)→ F (t)







1

n

n∑i=1

Fi (nt)→ F (t)



Service Process Fluid Limit

The service time of each customer is are independent identicallydistributed.

We haveSn(nt)

n→ µt

where Sn(nt) denotes the number of potential services completed intime nt, t > 0, and µ denotes the service rate.


Service Process Fluid Limit

The service time of each customer is are independent identicallydistributed.

We haveSn(nt)

n→ µt

where Sn(nt) denotes the number of potential services completed intime nt, t > 0, and µ denotes the service rate.


Fluid Limit of Net Input

The limiting net input process: Arrivals - potential service completions

X (t) = F (t),

for t < 0.

For t > 0,X (t) = F (t)− µt.


Fluid Limit of Net Input

The limiting net input process: Arrivals - potential service completions

X (t) = F (t),

for t < 0.

For t > 0,X (t) = F (t)− µt.


Queue Length Fluid Limit

The limiting queue length process

Q(t) = X (t) = F (t)

for t < 0.

For t > 0,Q(t) = X (t)− inf

0≤s≤t[X (s) ∧ 0]

where − inf0≤s≤t [X (s) ∧ 0] denotes the unused capacity by time t

If the queue has not emptied till time t > 0, then

Q̄(t) = F (t)− µt.




Q(t) = X (t) = F (t)

for t < 0.


0≤s≤t[X (s) ∧ 0]



Q̄(t) = F (t)− µt.




Q(t) = X (t) = F (t)

for t < 0.


0≤s≤t[X (s) ∧ 0]



Q̄(t) = F (t)− µt.


Queue Length Process


Queue Length Process: Server Fully Loaded


When to ‘Arrive’

Let WF (t) denote waiting time for arrival at time t when allother customers have an arrival profile F .

Arrival at time t incurs cost

CF (t) = αWF (t) + β(t + WF (t))

More generally, the expected cost incurred by a customer whoselects her arrival by sampling from probability distribution G is

CF (G ) =

∫ ∞−∞

(αWF (t) + β(t + WF (t))) dG (t) .


Nash Equilibrium

A multi-strategy {Gs(·), s ∈ [0, 1]} is a Nash equilibrium point if

(i) F (t) =∫ 1

0Gs(t)ds is well defined for each t, and

(ii) For any customer s ∈ [0, 1],

CF (Gs) ≤ CF (G̃ ), for every CDF G̃ .

That is, no customer s can improve his cost by modifying hisown arrival time distribution.

This corresponds to finding an arrival profile F such that CF (t) isconstant on its support and higher elsewhere.


Useful Insights

Let t∗ = inf{t ≥ 0 : F (t) < µt}.

In Nash Equilibrium, first time the server has spare capacity iswhen all customers are served. That is t∗ = 1/µ.

Similarly, in Nash equilibrium there can be no point masses in F .


Useful Insights

Let t∗ = inf{t ≥ 0 : F (t) < µt}.

In Nash Equilibrium, first time the server has spare capacity iswhen all customers are served. That is t∗ = 1/µ.

Similarly, in Nash equilibrium there can be no point masses in F .


Cost Function under Equilibrium

If F (·) denotes an equilibrium profile Then, W (t) equals

F (t)/µ− t.

The cost function CF (t) equals

β(t + W (t)) + αW (t) = (α + β)F (t)/µ− αt

For this to be independent of time F should be uniformlydistributed




F (t)/µ− t.


β(t + W (t)) + αW (t) = (α + β)F (t)/µ− αt





F (t)/µ− t.


β(t + W (t)) + αW (t) = (α + β)F (t)/µ− αt



Queue Length Process

Higher the β (time to service cost), higher the queue.


Socially Optimal Solution

The smallest value of waiting time is zero.

Total time to service is minimized if the server serves at thefastest possible rate. So the lower bound on average service timeis 1/(2µ) and on the overall cost is β/(2µ)

Easy to achieve. Price of anarchy equals two












Multi-Class Customers with Linear Costs

In equilibrium the different classes separate into disjoint intervalsand arrive in descending order of βi/αi .


Multi-Class Customers, Social optimal, POA

Class with higher β comes earlier. No queueing.

When all αi are equal, the POA equals 2

Let Gmax = maxi ,j G (i , j) and Gmin = mini ,j G (i , j), where

G (i , j) =(αi + αj) min{ βi

αi,βjαj}

2 min{βi , βj}.

Then2Gmin ≤ PoA ≤ 2Gmax.







αi,βjαj}

2 min{βi , βj}.








αi,βjαj}

2 min{βi , βj}.



Reducing Anarchy: Service Time Restrictions

Here (1− a) proportion is allowed to be served after time a/µ.PoA equals 2(a2 + (1− a)).Minimised to 3/2 at a=1/2.


Reducing Price of Anarchy

One generalization: m/n proportion of people come after time(n −m)/nµ, for m = 1, 2, 3, . . . , n − 1.

Easy to see that POLA now equals (n + 1)/n and converges to 1as n→∞.


Reducing PoA: Differential Pricing

Charge the customers that come in the first half p = β2µ

The queue reduces by half. Cafeteria gain equals β4µ.


If the amount charged is higher

Charge the customers that come in the first half p > β2µ

Cafeteria gains more.Its optimal p = 3

4βµ

and revenue 932βµ

.


If amount charged is lower

Better to err with lower prices than higher prices


Modified Cost Structure

Replace the cost

β(t + W (t)) + αW (t)

withβF (t) + αW (t).

The overall cost now equals

F (t)

µ(α + β̂)− αt,

where β̂ = βµ.

This differs from the previously analyzed cost function in that β̂replaces β. Price of Anarchy Remains 2



Replace the cost

β(t + W (t)) + αW (t)



F (t)

µ(α + β̂)− αt,

where β̂ = βµ.




Replace the cost

β(t + W (t)) + αW (t)



F (t)

µ(α + β̂)− αt,

where β̂ = βµ.



Numerical Results

Uniform arrivals, exponential service times

Normalized arrival time

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

-0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

N=10,000

N=1,000

N=500

N=100

N=50

N=10


Numerical Results

Deterministic arrivals, uniform (less variable) service times

Normalized arrival time

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

-0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

N=10,000

N=1,000

N=500

N=100

N=50

N=10


Non-linear Costs, Multi-class Customers

Adapted from Lindsay (2004). Waiting times for iso-cost curves.Upper envelope gives equilibrium schedule for the achievedcapacitiesComputational algorithm to solve for given capacities. Simpleuniqueness proof








Conclusions

We considered the queueing problem that may arise in settingssuch as concert halls, movie theaters, cafeterias etc.

The customers strategically selected their arrival timedistributions.

We developed a queueing framework and identified the fluidlimit. Fluid limits allow a great deal of tractability in analyzingthe strategic arrival problem faced by each customer.

We identified a Nash equilibrium strategy in multi-class settingwith linear costs and showed bounds on price of anarchy.

We discussed some ways to control price of anarchy

We discussed equilibrium strategies under non-linear costs andalgorithms to identify these.


Conclusions








Conclusions








Conclusions








Conclusions








Conclusions








The Concert/Cafeteria Queueing Problem A Game of Arrivals...Joint work with Rahul Jain (USC, LA) and Nahum Shimkin (Technion, Israel) ()Game of Arrivals Cambridge, June 16 3 / 30.

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