To jump the queue or wait my turn? - ERNETifcam/new_avenue/Slides/Seminars/Manjunath.pdfD. Manjunath...

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To jump the queue or wait my turn? D. Manjunath IIT Bombay January 14, 2014 D. Manjunath (IIT Bombay) To jump the queue or wait my turn? January 14, 2014 1 / 24

Transcript of To jump the queue or wait my turn? - ERNETifcam/new_avenue/Slides/Seminars/Manjunath.pdfD. Manjunath...

Page 1: To jump the queue or wait my turn? - ERNETifcam/new_avenue/Slides/Seminars/Manjunath.pdfD. Manjunath (IIT Bombay)To jump the queue or wait my turn?January 14, 2014 4 / 24 Adding value

To jump the queue or wait my turn?

D. Manjunath

IIT Bombay

January 14, 2014

D. Manjunath (IIT Bombay) To jump the queue or wait my turn? January 14, 2014 1 / 24

Page 2: To jump the queue or wait my turn? - ERNETifcam/new_avenue/Slides/Seminars/Manjunath.pdfD. Manjunath (IIT Bombay)To jump the queue or wait my turn?January 14, 2014 4 / 24 Adding value

Bidding for priorities

A single server queue has customers arriving according to ahomogeneous Poisson process of rate λ.

Service times are i.i.d. with distribution B(y) and mean 1µ ; Define

ρ := λµ .

There is no balking from the system; all arrivals receive service.

An arriving customer bids to pay X rupees for service.

Higher bids provide higher priorities with either preemptive resume ornon preemptive priority service.

Assume that the bids are i.i.d. and chosen according to a continuousdistribution B(x).

D. Manjunath (IIT Bombay) To jump the queue or wait my turn? January 14, 2014 2 / 24

Page 3: To jump the queue or wait my turn? - ERNETifcam/new_avenue/Slides/Seminars/Manjunath.pdfD. Manjunath (IIT Bombay)To jump the queue or wait my turn?January 14, 2014 4 / 24 Adding value

Bidding for priorities

A single server queue has customers arriving according to ahomogeneous Poisson process of rate λ.

Service times are i.i.d. with distribution B(y) and mean 1µ ; Define

ρ := λµ .

There is no balking from the system; all arrivals receive service.

An arriving customer bids to pay X rupees for service.

Higher bids provide higher priorities with either preemptive resume ornon preemptive priority service.

Assume that the bids are i.i.d. and chosen according to a continuousdistribution B(x).

D. Manjunath (IIT Bombay) To jump the queue or wait my turn? January 14, 2014 2 / 24

Page 4: To jump the queue or wait my turn? - ERNETifcam/new_avenue/Slides/Seminars/Manjunath.pdfD. Manjunath (IIT Bombay)To jump the queue or wait my turn?January 14, 2014 4 / 24 Adding value

Bidding for priorities

A single server queue has customers arriving according to ahomogeneous Poisson process of rate λ.

Service times are i.i.d. with distribution B(y) and mean 1µ ; Define

ρ := λµ .

There is no balking from the system; all arrivals receive service.

An arriving customer bids to pay X rupees for service.

Higher bids provide higher priorities with either preemptive resume ornon preemptive priority service.

Assume that the bids are i.i.d. and chosen according to a continuousdistribution B(x).

D. Manjunath (IIT Bombay) To jump the queue or wait my turn? January 14, 2014 2 / 24

Page 5: To jump the queue or wait my turn? - ERNETifcam/new_avenue/Slides/Seminars/Manjunath.pdfD. Manjunath (IIT Bombay)To jump the queue or wait my turn?January 14, 2014 4 / 24 Adding value

Bidding for priorities

A single server queue has customers arriving according to ahomogeneous Poisson process of rate λ.

Service times are i.i.d. with distribution B(y) and mean 1µ ; Define

ρ := λµ .

There is no balking from the system; all arrivals receive service.

An arriving customer bids to pay X rupees for service.

Higher bids provide higher priorities with either preemptive resume ornon preemptive priority service.

Assume that the bids are i.i.d. and chosen according to a continuousdistribution B(x).

D. Manjunath (IIT Bombay) To jump the queue or wait my turn? January 14, 2014 2 / 24

Page 6: To jump the queue or wait my turn? - ERNETifcam/new_avenue/Slides/Seminars/Manjunath.pdfD. Manjunath (IIT Bombay)To jump the queue or wait my turn?January 14, 2014 4 / 24 Adding value

Bidding for priorities

A single server queue has customers arriving according to ahomogeneous Poisson process of rate λ.

Service times are i.i.d. with distribution B(y) and mean 1µ ; Define

ρ := λµ .

There is no balking from the system; all arrivals receive service.

An arriving customer bids to pay X rupees for service.

Higher bids provide higher priorities with either preemptive resume ornon preemptive priority service.

Assume that the bids are i.i.d. and chosen according to a continuousdistribution B(x).

D. Manjunath (IIT Bombay) To jump the queue or wait my turn? January 14, 2014 2 / 24

Page 7: To jump the queue or wait my turn? - ERNETifcam/new_avenue/Slides/Seminars/Manjunath.pdfD. Manjunath (IIT Bombay)To jump the queue or wait my turn?January 14, 2014 4 / 24 Adding value

Bidding for priorities

A single server queue has customers arriving according to ahomogeneous Poisson process of rate λ.

Service times are i.i.d. with distribution B(y) and mean 1µ ; Define

ρ := λµ .

There is no balking from the system; all arrivals receive service.

An arriving customer bids to pay X rupees for service.

Higher bids provide higher priorities with either preemptive resume ornon preemptive priority service.

Assume that the bids are i.i.d. and chosen according to a continuousdistribution B(x).

D. Manjunath (IIT Bombay) To jump the queue or wait my turn? January 14, 2014 2 / 24

Page 8: To jump the queue or wait my turn? - ERNETifcam/new_avenue/Slides/Seminars/Manjunath.pdfD. Manjunath (IIT Bombay)To jump the queue or wait my turn?January 14, 2014 4 / 24 Adding value

Bidding for prioritiesSojourn time of customer bidding x , W (x), has three components.

Its own service time which has mean 1µ .

Waiting caused by those already in system and having bids higher thanx . From Little’s law and PASTA property, the expectation will be

1

µ

[∫ ∞

x

W (y) (λdB(y))

].

Waiting time due to those that come after its arrival but during its stayin the system and bid higher than x . This is

1

µW (x)

[∫ ∞

x

λdB(y)

].

Adding the three terms, and after some work, we obtain

W (x) =W0

(1− ρ+ ρB(x))2

where W0 is mean residual service time of customer in service for nonpreemptive priority service and 1/µ for preemptive priority service.

D. Manjunath (IIT Bombay) To jump the queue or wait my turn? January 14, 2014 3 / 24

Page 9: To jump the queue or wait my turn? - ERNETifcam/new_avenue/Slides/Seminars/Manjunath.pdfD. Manjunath (IIT Bombay)To jump the queue or wait my turn?January 14, 2014 4 / 24 Adding value

Bidding for prioritiesSojourn time of customer bidding x , W (x), has three components.

Its own service time which has mean 1µ .

Waiting caused by those already in system and having bids higher thanx . From Little’s law and PASTA property, the expectation will be

1

µ

[∫ ∞

x

W (y) (λdB(y))

].

Waiting time due to those that come after its arrival but during its stayin the system and bid higher than x . This is

1

µW (x)

[∫ ∞

x

λdB(y)

].

Adding the three terms, and after some work, we obtain

W (x) =W0

(1− ρ+ ρB(x))2

where W0 is mean residual service time of customer in service for nonpreemptive priority service and 1/µ for preemptive priority service.

D. Manjunath (IIT Bombay) To jump the queue or wait my turn? January 14, 2014 3 / 24

Page 10: To jump the queue or wait my turn? - ERNETifcam/new_avenue/Slides/Seminars/Manjunath.pdfD. Manjunath (IIT Bombay)To jump the queue or wait my turn?January 14, 2014 4 / 24 Adding value

Bidding for prioritiesSojourn time of customer bidding x , W (x), has three components.

Its own service time which has mean 1µ .

Waiting caused by those already in system and having bids higher thanx . From Little’s law and PASTA property, the expectation will be

1

µ

[∫ ∞

x

W (y) (λdB(y))

].

Waiting time due to those that come after its arrival but during its stayin the system and bid higher than x . This is

1

µW (x)

[∫ ∞

x

λdB(y)

].

Adding the three terms, and after some work, we obtain

W (x) =W0

(1− ρ+ ρB(x))2

where W0 is mean residual service time of customer in service for nonpreemptive priority service and 1/µ for preemptive priority service.

D. Manjunath (IIT Bombay) To jump the queue or wait my turn? January 14, 2014 3 / 24

Page 11: To jump the queue or wait my turn? - ERNETifcam/new_avenue/Slides/Seminars/Manjunath.pdfD. Manjunath (IIT Bombay)To jump the queue or wait my turn?January 14, 2014 4 / 24 Adding value

Bidding for prioritiesSojourn time of customer bidding x , W (x), has three components.

Its own service time which has mean 1µ .

Waiting caused by those already in system and having bids higher thanx . From Little’s law and PASTA property, the expectation will be

1

µ

[∫ ∞

x

W (y) (λdB(y))

].

Waiting time due to those that come after its arrival but during its stayin the system and bid higher than x . This is

1

µW (x)

[∫ ∞

x

λdB(y)

].

Adding the three terms, and after some work, we obtain

W (x) =W0

(1− ρ+ ρB(x))2

where W0 is mean residual service time of customer in service for nonpreemptive priority service and 1/µ for preemptive priority service.

D. Manjunath (IIT Bombay) To jump the queue or wait my turn? January 14, 2014 3 / 24

Page 12: To jump the queue or wait my turn? - ERNETifcam/new_avenue/Slides/Seminars/Manjunath.pdfD. Manjunath (IIT Bombay)To jump the queue or wait my turn?January 14, 2014 4 / 24 Adding value

Bidding for prioritiesSojourn time of customer bidding x , W (x), has three components.

Its own service time which has mean 1µ .

Waiting caused by those already in system and having bids higher thanx . From Little’s law and PASTA property, the expectation will be

1

µ

[∫ ∞

x

W (y) (λdB(y))

].

Waiting time due to those that come after its arrival but during its stayin the system and bid higher than x . This is

1

µW (x)

[∫ ∞

x

λdB(y)

].

Adding the three terms, and after some work, we obtain

W (x) =W0

(1− ρ+ ρB(x))2

where W0 is mean residual service time of customer in service for nonpreemptive priority service and 1/µ for preemptive priority service.

D. Manjunath (IIT Bombay) To jump the queue or wait my turn? January 14, 2014 3 / 24

Page 13: To jump the queue or wait my turn? - ERNETifcam/new_avenue/Slides/Seminars/Manjunath.pdfD. Manjunath (IIT Bombay)To jump the queue or wait my turn?January 14, 2014 4 / 24 Adding value

Adding value to time

Now let each customer have a non negative delay sensitivityparameter V . Specifically, let V be cost per unit waiting time.

Assume that V of an arriving customer is i.i.d. with distributionF (v), F (v) has a density with support in (a, b), i.e.,

F (v) =

{0 0 < a < v

1 v > b

F (v) is the delay sensitivity profile of the customer population.

Let bidding function X (v) denote bid of customers of sensitivity v .

Following preceding notation, total cost for customer of sensitivity v is

C (v) := X (v) + vW (X (v)).

D. Manjunath (IIT Bombay) To jump the queue or wait my turn? January 14, 2014 4 / 24

Page 14: To jump the queue or wait my turn? - ERNETifcam/new_avenue/Slides/Seminars/Manjunath.pdfD. Manjunath (IIT Bombay)To jump the queue or wait my turn?January 14, 2014 4 / 24 Adding value

Adding value to time

Now let each customer have a non negative delay sensitivityparameter V . Specifically, let V be cost per unit waiting time.

Assume that V of an arriving customer is i.i.d. with distributionF (v), F (v) has a density with support in (a, b), i.e.,

F (v) =

{0 0 < a < v

1 v > b

F (v) is the delay sensitivity profile of the customer population.

Let bidding function X (v) denote bid of customers of sensitivity v .

Following preceding notation, total cost for customer of sensitivity v is

C (v) := X (v) + vW (X (v)).

D. Manjunath (IIT Bombay) To jump the queue or wait my turn? January 14, 2014 4 / 24

Page 15: To jump the queue or wait my turn? - ERNETifcam/new_avenue/Slides/Seminars/Manjunath.pdfD. Manjunath (IIT Bombay)To jump the queue or wait my turn?January 14, 2014 4 / 24 Adding value

Adding value to time

Now let each customer have a non negative delay sensitivityparameter V . Specifically, let V be cost per unit waiting time.

Assume that V of an arriving customer is i.i.d. with distributionF (v), F (v) has a density with support in (a, b), i.e.,

F (v) =

{0 0 < a < v

1 v > b

F (v) is the delay sensitivity profile of the customer population.

Let bidding function X (v) denote bid of customers of sensitivity v .

Following preceding notation, total cost for customer of sensitivity v is

C (v) := X (v) + vW (X (v)).

D. Manjunath (IIT Bombay) To jump the queue or wait my turn? January 14, 2014 4 / 24

Page 16: To jump the queue or wait my turn? - ERNETifcam/new_avenue/Slides/Seminars/Manjunath.pdfD. Manjunath (IIT Bombay)To jump the queue or wait my turn?January 14, 2014 4 / 24 Adding value

Adding value to time

Now let each customer have a non negative delay sensitivityparameter V . Specifically, let V be cost per unit waiting time.

Assume that V of an arriving customer is i.i.d. with distributionF (v), F (v) has a density with support in (a, b), i.e.,

F (v) =

{0 0 < a < v

1 v > b

F (v) is the delay sensitivity profile of the customer population.

Let bidding function X (v) denote bid of customers of sensitivity v .

Following preceding notation, total cost for customer of sensitivity v is

C (v) := X (v) + vW (X (v)).

D. Manjunath (IIT Bombay) To jump the queue or wait my turn? January 14, 2014 4 / 24

Page 17: To jump the queue or wait my turn? - ERNETifcam/new_avenue/Slides/Seminars/Manjunath.pdfD. Manjunath (IIT Bombay)To jump the queue or wait my turn?January 14, 2014 4 / 24 Adding value

Adding value to time (contd.)

Let BX (x) be the distribution of the bids under bidding policy X (v)and W X (x) be the waiting time of a customers that bid x underpolicy X .

Now assume customers are strategic—X (v) minimises total costC (v).

Bidding function X (v) is said to be in equilibrium if no individualcustomer has an incentive to deviate from it.

If customers follow an equilibrium bidding policy, then C (v) isminimum for each v , and a customer of sensitivity v cannotunilaterally deviate from the equilibrium X (v) and lower its C (v).

D. Manjunath (IIT Bombay) To jump the queue or wait my turn? January 14, 2014 5 / 24

Page 18: To jump the queue or wait my turn? - ERNETifcam/new_avenue/Slides/Seminars/Manjunath.pdfD. Manjunath (IIT Bombay)To jump the queue or wait my turn?January 14, 2014 4 / 24 Adding value

Adding value to time (contd.)

Let BX (x) be the distribution of the bids under bidding policy X (v)and W X (x) be the waiting time of a customers that bid x underpolicy X .

Now assume customers are strategic—X (v) minimises total costC (v).

Bidding function X (v) is said to be in equilibrium if no individualcustomer has an incentive to deviate from it.

If customers follow an equilibrium bidding policy, then C (v) isminimum for each v , and a customer of sensitivity v cannotunilaterally deviate from the equilibrium X (v) and lower its C (v).

D. Manjunath (IIT Bombay) To jump the queue or wait my turn? January 14, 2014 5 / 24

Page 19: To jump the queue or wait my turn? - ERNETifcam/new_avenue/Slides/Seminars/Manjunath.pdfD. Manjunath (IIT Bombay)To jump the queue or wait my turn?January 14, 2014 4 / 24 Adding value

Adding value to time (contd.)

Let BX (x) be the distribution of the bids under bidding policy X (v)and W X (x) be the waiting time of a customers that bid x underpolicy X .

Now assume customers are strategic—X (v) minimises total costC (v).

Bidding function X (v) is said to be in equilibrium if no individualcustomer has an incentive to deviate from it.

If customers follow an equilibrium bidding policy, then C (v) isminimum for each v , and a customer of sensitivity v cannotunilaterally deviate from the equilibrium X (v) and lower its C (v).

D. Manjunath (IIT Bombay) To jump the queue or wait my turn? January 14, 2014 5 / 24

Page 20: To jump the queue or wait my turn? - ERNETifcam/new_avenue/Slides/Seminars/Manjunath.pdfD. Manjunath (IIT Bombay)To jump the queue or wait my turn?January 14, 2014 4 / 24 Adding value

Adding value to time (contd.)

Let BX (x) be the distribution of the bids under bidding policy X (v)and W X (x) be the waiting time of a customers that bid x underpolicy X .

Now assume customers are strategic—X (v) minimises total costC (v).

Bidding function X (v) is said to be in equilibrium if no individualcustomer has an incentive to deviate from it.

If customers follow an equilibrium bidding policy, then C (v) isminimum for each v , and a customer of sensitivity v cannotunilaterally deviate from the equilibrium X (v) and lower its C (v).

D. Manjunath (IIT Bombay) To jump the queue or wait my turn? January 14, 2014 5 / 24

Page 21: To jump the queue or wait my turn? - ERNETifcam/new_avenue/Slides/Seminars/Manjunath.pdfD. Manjunath (IIT Bombay)To jump the queue or wait my turn?January 14, 2014 4 / 24 Adding value

Characterising stable policies

Lemma

If X (v) is a stable bidding function, BX (x) is continuous and strictlyincreasing in x .

Property

There exists a continuous X (v) that is a stable bidding policy. For such apolicy, the following hold.

1 W X (x) is strictly decreasing in x .

2 CX (v) is continuous and strictly increasing. Further, it is concave in

v and dC(v)dv = W (x).

3 If v1 < v2 then X (v1) < X (v2).

4 B(X (v)) = F (v).

D. Manjunath (IIT Bombay) To jump the queue or wait my turn? January 14, 2014 6 / 24

Page 22: To jump the queue or wait my turn? - ERNETifcam/new_avenue/Slides/Seminars/Manjunath.pdfD. Manjunath (IIT Bombay)To jump the queue or wait my turn?January 14, 2014 4 / 24 Adding value

Characterising stable policies

Lemma

If X (v) is a stable bidding function, BX (x) is continuous and strictlyincreasing in x .

Property

There exists a continuous X (v) that is a stable bidding policy. For such apolicy, the following hold.

1 W X (x) is strictly decreasing in x .

2 CX (v) is continuous and strictly increasing. Further, it is concave in

v and dC(v)dv = W (x).

3 If v1 < v2 then X (v1) < X (v2).

4 B(X (v)) = F (v).

D. Manjunath (IIT Bombay) To jump the queue or wait my turn? January 14, 2014 6 / 24

Page 23: To jump the queue or wait my turn? - ERNETifcam/new_avenue/Slides/Seminars/Manjunath.pdfD. Manjunath (IIT Bombay)To jump the queue or wait my turn?January 14, 2014 4 / 24 Adding value

Characterising stable policies

Lemma

If X (v) is a stable bidding function, BX (x) is continuous and strictlyincreasing in x .

Property

There exists a continuous X (v) that is a stable bidding policy. For such apolicy, the following hold.

1 W X (x) is strictly decreasing in x .

2 CX (v) is continuous and strictly increasing. Further, it is concave in

v and dC(v)dv = W (x).

3 If v1 < v2 then X (v1) < X (v2).

4 B(X (v)) = F (v).

D. Manjunath (IIT Bombay) To jump the queue or wait my turn? January 14, 2014 6 / 24

Page 24: To jump the queue or wait my turn? - ERNETifcam/new_avenue/Slides/Seminars/Manjunath.pdfD. Manjunath (IIT Bombay)To jump the queue or wait my turn?January 14, 2014 4 / 24 Adding value

Characterising stable policies

Lemma

If X (v) is a stable bidding function, BX (x) is continuous and strictlyincreasing in x .

Property

There exists a continuous X (v) that is a stable bidding policy. For such apolicy, the following hold.

1 W X (x) is strictly decreasing in x .

2 CX (v) is continuous and strictly increasing. Further, it is concave in

v and dC(v)dv = W (x).

3 If v1 < v2 then X (v1) < X (v2).

4 B(X (v)) = F (v).

D. Manjunath (IIT Bombay) To jump the queue or wait my turn? January 14, 2014 6 / 24

Page 25: To jump the queue or wait my turn? - ERNETifcam/new_avenue/Slides/Seminars/Manjunath.pdfD. Manjunath (IIT Bombay)To jump the queue or wait my turn?January 14, 2014 4 / 24 Adding value

Characterising stable policies

Lemma

If X (v) is a stable bidding function, BX (x) is continuous and strictlyincreasing in x .

Property

There exists a continuous X (v) that is a stable bidding policy. For such apolicy, the following hold.

1 W X (x) is strictly decreasing in x .

2 CX (v) is continuous and strictly increasing. Further, it is concave in

v and dC(v)dv = W (x).

3 If v1 < v2 then X (v1) < X (v2).

4 B(X (v)) = F (v).

D. Manjunath (IIT Bombay) To jump the queue or wait my turn? January 14, 2014 6 / 24

Page 26: To jump the queue or wait my turn? - ERNETifcam/new_avenue/Slides/Seminars/Manjunath.pdfD. Manjunath (IIT Bombay)To jump the queue or wait my turn?January 14, 2014 4 / 24 Adding value

Characterising stable policies

Lemma

If X (v) is a stable bidding function, BX (x) is continuous and strictlyincreasing in x .

Property

There exists a continuous X (v) that is a stable bidding policy. For such apolicy, the following hold.

1 W X (x) is strictly decreasing in x .

2 CX (v) is continuous and strictly increasing. Further, it is concave in

v and dC(v)dv = W (x).

3 If v1 < v2 then X (v1) < X (v2).

4 B(X (v)) = F (v).

D. Manjunath (IIT Bombay) To jump the queue or wait my turn? January 14, 2014 6 / 24

Page 27: To jump the queue or wait my turn? - ERNETifcam/new_avenue/Slides/Seminars/Manjunath.pdfD. Manjunath (IIT Bombay)To jump the queue or wait my turn?January 14, 2014 4 / 24 Adding value

Revenue to the server

For v ∈ (a, b),

X (v) =

∫ v

0

2λW0y

(1− ρ− ρF (y))3dF (y)

is a stable bidding policy. W0 is as before—mean residual service timefor non preemptive priority policy and 1/mu for preemptive prioiritypolicy.

Fixing the arrival rate and the delay sensitivity function F (v), abidding policy X (v) will yield an expected revenue rate of

RX (λ,F (v)) = λ

∫v

X (v)dF (v).

D. Manjunath (IIT Bombay) To jump the queue or wait my turn? January 14, 2014 7 / 24

Page 28: To jump the queue or wait my turn? - ERNETifcam/new_avenue/Slides/Seminars/Manjunath.pdfD. Manjunath (IIT Bombay)To jump the queue or wait my turn?January 14, 2014 4 / 24 Adding value

Revenue to the server

For v ∈ (a, b),

X (v) =

∫ v

0

2λW0y

(1− ρ− ρF (y))3dF (y)

is a stable bidding policy. W0 is as before—mean residual service timefor non preemptive priority policy and 1/mu for preemptive prioiritypolicy.

Fixing the arrival rate and the delay sensitivity function F (v), abidding policy X (v) will yield an expected revenue rate of

RX (λ,F (v)) = λ

∫v

X (v)dF (v).

D. Manjunath (IIT Bombay) To jump the queue or wait my turn? January 14, 2014 7 / 24

Page 29: To jump the queue or wait my turn? - ERNETifcam/new_avenue/Slides/Seminars/Manjunath.pdfD. Manjunath (IIT Bombay)To jump the queue or wait my turn?January 14, 2014 4 / 24 Adding value

Properties of the Revenue Function

1 Let F1(v) be a sensitivity profile and F2(v) = F1(v − v0) wherev0 > 0. Then R(λ,F1(v)) < R(λ,F2(v)).

2 If λ1 < λ2 then R(λ1,F (v)) < R(λ2,F (v))

3 Consider the scaling of the sensitivity profile, i.e., consider F (αv) forα > 0. Then R(λ,F (v)) < R(λ,F (αv)) when α > 1.

4 Let a < a1 < b, and define

F1(v) =

0 v < a1F (v)∫ b

a1dF (x)

v ≥ a1

Then R(λ,F (v)) < R(λ,F1(v)).

D. Manjunath (IIT Bombay) To jump the queue or wait my turn? January 14, 2014 8 / 24

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Properties of the Revenue Function

1 Let F1(v) be a sensitivity profile and F2(v) = F1(v − v0) wherev0 > 0. Then R(λ,F1(v)) < R(λ,F2(v)).

2 If λ1 < λ2 then R(λ1,F (v)) < R(λ2,F (v))

3 Consider the scaling of the sensitivity profile, i.e., consider F (αv) forα > 0. Then R(λ,F (v)) < R(λ,F (αv)) when α > 1.

4 Let a < a1 < b, and define

F1(v) =

0 v < a1F (v)∫ b

a1dF (x)

v ≥ a1

Then R(λ,F (v)) < R(λ,F1(v)).

D. Manjunath (IIT Bombay) To jump the queue or wait my turn? January 14, 2014 8 / 24

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Properties of the Revenue Function

1 Let F1(v) be a sensitivity profile and F2(v) = F1(v − v0) wherev0 > 0. Then R(λ,F1(v)) < R(λ,F2(v)).

2 If λ1 < λ2 then R(λ1,F (v)) < R(λ2,F (v))

3 Consider the scaling of the sensitivity profile, i.e., consider F (αv) forα > 0. Then R(λ,F (v)) < R(λ,F (αv)) when α > 1.

4 Let a < a1 < b, and define

F1(v) =

0 v < a1F (v)∫ b

a1dF (x)

v ≥ a1

Then R(λ,F (v)) < R(λ,F1(v)).

D. Manjunath (IIT Bombay) To jump the queue or wait my turn? January 14, 2014 8 / 24

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Properties of the Revenue Function

1 Let F1(v) be a sensitivity profile and F2(v) = F1(v − v0) wherev0 > 0. Then R(λ,F1(v)) < R(λ,F2(v)).

2 If λ1 < λ2 then R(λ1,F (v)) < R(λ2,F (v))

3 Consider the scaling of the sensitivity profile, i.e., consider F (αv) forα > 0. Then R(λ,F (v)) < R(λ,F (αv)) when α > 1.

4 Let a < a1 < b, and define

F1(v) =

0 v < a1F (v)∫ b

a1dF (x)

v ≥ a1

Then R(λ,F (v)) < R(λ,F1(v)).

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Introducing a Second Queue

To the bidding queue, add a parallel FIFO queue that has service rateµ1 while the bidding queue has service rate µ2.

Thus arriving customers make two decisions—(1) which queue tojoin, and (2) if joining the bidding queue, what should be the bid.

There is no balking.

If a customer decides to receive service from the FIFO queue, hereceives a reward of M rupees.

Like before, we assume oblivious routing and oblivious bidding.

Customers are strategic—each one minimises its individual costfunction.

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Introducing a Second Queue

To the bidding queue, add a parallel FIFO queue that has service rateµ1 while the bidding queue has service rate µ2.

Thus arriving customers make two decisions—(1) which queue tojoin, and (2) if joining the bidding queue, what should be the bid.

There is no balking.

If a customer decides to receive service from the FIFO queue, hereceives a reward of M rupees.

Like before, we assume oblivious routing and oblivious bidding.

Customers are strategic—each one minimises its individual costfunction.

D. Manjunath (IIT Bombay) To jump the queue or wait my turn? January 14, 2014 9 / 24

Page 35: To jump the queue or wait my turn? - ERNETifcam/new_avenue/Slides/Seminars/Manjunath.pdfD. Manjunath (IIT Bombay)To jump the queue or wait my turn?January 14, 2014 4 / 24 Adding value

Introducing a Second Queue

To the bidding queue, add a parallel FIFO queue that has service rateµ1 while the bidding queue has service rate µ2.

Thus arriving customers make two decisions—(1) which queue tojoin, and (2) if joining the bidding queue, what should be the bid.

There is no balking.

If a customer decides to receive service from the FIFO queue, hereceives a reward of M rupees.

Like before, we assume oblivious routing and oblivious bidding.

Customers are strategic—each one minimises its individual costfunction.

D. Manjunath (IIT Bombay) To jump the queue or wait my turn? January 14, 2014 9 / 24

Page 36: To jump the queue or wait my turn? - ERNETifcam/new_avenue/Slides/Seminars/Manjunath.pdfD. Manjunath (IIT Bombay)To jump the queue or wait my turn?January 14, 2014 4 / 24 Adding value

Introducing a Second Queue

To the bidding queue, add a parallel FIFO queue that has service rateµ1 while the bidding queue has service rate µ2.

Thus arriving customers make two decisions—(1) which queue tojoin, and (2) if joining the bidding queue, what should be the bid.

There is no balking.

If a customer decides to receive service from the FIFO queue, hereceives a reward of M rupees.

Like before, we assume oblivious routing and oblivious bidding.

Customers are strategic—each one minimises its individual costfunction.

D. Manjunath (IIT Bombay) To jump the queue or wait my turn? January 14, 2014 9 / 24

Page 37: To jump the queue or wait my turn? - ERNETifcam/new_avenue/Slides/Seminars/Manjunath.pdfD. Manjunath (IIT Bombay)To jump the queue or wait my turn?January 14, 2014 4 / 24 Adding value

Introducing a Second Queue

To the bidding queue, add a parallel FIFO queue that has service rateµ1 while the bidding queue has service rate µ2.

Thus arriving customers make two decisions—(1) which queue tojoin, and (2) if joining the bidding queue, what should be the bid.

There is no balking.

If a customer decides to receive service from the FIFO queue, hereceives a reward of M rupees.

Like before, we assume oblivious routing and oblivious bidding.

Customers are strategic—each one minimises its individual costfunction.

D. Manjunath (IIT Bombay) To jump the queue or wait my turn? January 14, 2014 9 / 24

Page 38: To jump the queue or wait my turn? - ERNETifcam/new_avenue/Slides/Seminars/Manjunath.pdfD. Manjunath (IIT Bombay)To jump the queue or wait my turn?January 14, 2014 4 / 24 Adding value

Introducing a Second Queue

To the bidding queue, add a parallel FIFO queue that has service rateµ1 while the bidding queue has service rate µ2.

Thus arriving customers make two decisions—(1) which queue tojoin, and (2) if joining the bidding queue, what should be the bid.

There is no balking.

If a customer decides to receive service from the FIFO queue, hereceives a reward of M rupees.

Like before, we assume oblivious routing and oblivious bidding.

Customers are strategic—each one minimises its individual costfunction.

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More Notation

An arrival with sensitivity v has the following decisions

Which queue? Chooses FIFO queue with probability p(v).p(v) : <+ → [0, 1] is called the routing function.What price? Bid X (v) to be paid if it joins bidding queue;X (v) : <+ → <+ is the bidding function.S(v) := (p(v),X (v)) is strategy of customer of class v .

S(v) is a stable strategy if no customer has an incentive to deviate.

λ1 = λ∫∞0 p(v)dF (v) is the arrival rate to the FIFO queue

λ2 = λ− λ1 is the arrival rate to the bidding queueρ1 = λ1/µ1 and ρ2 = λ2/µ2.

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More Notation

An arrival with sensitivity v has the following decisions

Which queue? Chooses FIFO queue with probability p(v).p(v) : <+ → [0, 1] is called the routing function.

What price? Bid X (v) to be paid if it joins bidding queue;X (v) : <+ → <+ is the bidding function.S(v) := (p(v),X (v)) is strategy of customer of class v .

S(v) is a stable strategy if no customer has an incentive to deviate.

λ1 = λ∫∞0 p(v)dF (v) is the arrival rate to the FIFO queue

λ2 = λ− λ1 is the arrival rate to the bidding queueρ1 = λ1/µ1 and ρ2 = λ2/µ2.

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More Notation

An arrival with sensitivity v has the following decisions

Which queue? Chooses FIFO queue with probability p(v).p(v) : <+ → [0, 1] is called the routing function.What price? Bid X (v) to be paid if it joins bidding queue;X (v) : <+ → <+ is the bidding function.

S(v) := (p(v),X (v)) is strategy of customer of class v .

S(v) is a stable strategy if no customer has an incentive to deviate.

λ1 = λ∫∞0 p(v)dF (v) is the arrival rate to the FIFO queue

λ2 = λ− λ1 is the arrival rate to the bidding queueρ1 = λ1/µ1 and ρ2 = λ2/µ2.

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More Notation

An arrival with sensitivity v has the following decisions

Which queue? Chooses FIFO queue with probability p(v).p(v) : <+ → [0, 1] is called the routing function.What price? Bid X (v) to be paid if it joins bidding queue;X (v) : <+ → <+ is the bidding function.S(v) := (p(v),X (v)) is strategy of customer of class v .

S(v) is a stable strategy if no customer has an incentive to deviate.

λ1 = λ∫∞0 p(v)dF (v) is the arrival rate to the FIFO queue

λ2 = λ− λ1 is the arrival rate to the bidding queueρ1 = λ1/µ1 and ρ2 = λ2/µ2.

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More Notation

An arrival with sensitivity v has the following decisions

Which queue? Chooses FIFO queue with probability p(v).p(v) : <+ → [0, 1] is called the routing function.What price? Bid X (v) to be paid if it joins bidding queue;X (v) : <+ → <+ is the bidding function.S(v) := (p(v),X (v)) is strategy of customer of class v .

S(v) is a stable strategy if no customer has an incentive to deviate.

λ1 = λ∫∞0 p(v)dF (v) is the arrival rate to the FIFO queue

λ2 = λ− λ1 is the arrival rate to the bidding queueρ1 = λ1/µ1 and ρ2 = λ2/µ2.

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More Notation

An arrival with sensitivity v has the following decisions

Which queue? Chooses FIFO queue with probability p(v).p(v) : <+ → [0, 1] is called the routing function.What price? Bid X (v) to be paid if it joins bidding queue;X (v) : <+ → <+ is the bidding function.S(v) := (p(v),X (v)) is strategy of customer of class v .

S(v) is a stable strategy if no customer has an incentive to deviate.

λ1 = λ∫∞0 p(v)dF (v) is the arrival rate to the FIFO queue

λ2 = λ− λ1 is the arrival rate to the bidding queueρ1 = λ1/µ1 and ρ2 = λ2/µ2.

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Remarks

Delay in FIFO queue is independent of the class; W1 is theexpectation.

Delay in bidding queue is a function of v ; W2(v) is the expectation.

Let XE (v) be a equilibrium strategy; equilibrium will be Wardrop andthe following will be true.

p(v) = 1 implies vW1 < X (v) + W2(XE (v)).p(v) = 0 implies vW1 > X (v) + W2(XE (v)).0 < p(v) < 1 implies vW1 −M = XE (v) + W2(XE (v)).

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Remarks

Delay in FIFO queue is independent of the class; W1 is theexpectation.

Delay in bidding queue is a function of v ; W2(v) is the expectation.

Let XE (v) be a equilibrium strategy; equilibrium will be Wardrop andthe following will be true.

p(v) = 1 implies vW1 < X (v) + W2(XE (v)).p(v) = 0 implies vW1 > X (v) + W2(XE (v)).0 < p(v) < 1 implies vW1 −M = XE (v) + W2(XE (v)).

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Remarks

Delay in FIFO queue is independent of the class; W1 is theexpectation.

Delay in bidding queue is a function of v ; W2(v) is the expectation.

Let XE (v) be a equilibrium strategy; equilibrium will be Wardrop andthe following will be true.

p(v) = 1 implies vW1 < X (v) + W2(XE (v)).p(v) = 0 implies vW1 > X (v) + W2(XE (v)).0 < p(v) < 1 implies vW1 −M = XE (v) + W2(XE (v)).

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Remarks

Delay in FIFO queue is independent of the class; W1 is theexpectation.

Delay in bidding queue is a function of v ; W2(v) is the expectation.

Let XE (v) be a equilibrium strategy; equilibrium will be Wardrop andthe following will be true.

p(v) = 1 implies vW1 < X (v) + W2(XE (v)).

p(v) = 0 implies vW1 > X (v) + W2(XE (v)).0 < p(v) < 1 implies vW1 −M = XE (v) + W2(XE (v)).

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Remarks

Delay in FIFO queue is independent of the class; W1 is theexpectation.

Delay in bidding queue is a function of v ; W2(v) is the expectation.

Let XE (v) be a equilibrium strategy; equilibrium will be Wardrop andthe following will be true.

p(v) = 1 implies vW1 < X (v) + W2(XE (v)).p(v) = 0 implies vW1 > X (v) + W2(XE (v)).

0 < p(v) < 1 implies vW1 −M = XE (v) + W2(XE (v)).

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Remarks

Delay in FIFO queue is independent of the class; W1 is theexpectation.

Delay in bidding queue is a function of v ; W2(v) is the expectation.

Let XE (v) be a equilibrium strategy; equilibrium will be Wardrop andthe following will be true.

p(v) = 1 implies vW1 < X (v) + W2(XE (v)).p(v) = 0 implies vW1 > X (v) + W2(XE (v)).0 < p(v) < 1 implies vW1 −M = XE (v) + W2(XE (v)).

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A Theorem

For 0 ≤ t ≤ 1 and a ≤ v1 ≤ b define

pE (v) =

0 for v > v1,

t for v = v1,

1 for v < v1.

F1(v) :=

0 v ≤ v1

F (v)∫ bv1

dF (v)v1 ≤ v ≤ b,

1 v > b.

XE (v) =

∫ v

0

2λ2W0y

(1− ρ2 − ρ2F1(y))3dF1(y)

Let W1 be expected delay in FIFO queue with arrival rate λ1 andW2(v) expected delay in bidding queue with bidding function XE (v).

Let v1 satisfy v1W2(v1) = v1W1 −M.

(pE (v),XE (v)) is a stable policy with v1 as above.

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A Theorem

For 0 ≤ t ≤ 1 and a ≤ v1 ≤ b define

pE (v) =

0 for v > v1,

t for v = v1,

1 for v < v1.

F1(v) :=

0 v ≤ v1

F (v)∫ bv1

dF (v)v1 ≤ v ≤ b,

1 v > b.

XE (v) =

∫ v

0

2λ2W0y

(1− ρ2 − ρ2F1(y))3dF1(y)

Let W1 be expected delay in FIFO queue with arrival rate λ1 andW2(v) expected delay in bidding queue with bidding function XE (v).

Let v1 satisfy v1W2(v1) = v1W1 −M.

(pE (v),XE (v)) is a stable policy with v1 as above.

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A Theorem

For 0 ≤ t ≤ 1 and a ≤ v1 ≤ b define

pE (v) =

0 for v > v1,

t for v = v1,

1 for v < v1.

F1(v) :=

0 v ≤ v1

F (v)∫ bv1

dF (v)v1 ≤ v ≤ b,

1 v > b.

XE (v) =

∫ v

0

2λ2W0y

(1− ρ2 − ρ2F1(y))3dF1(y)

Let W1 be expected delay in FIFO queue with arrival rate λ1 andW2(v) expected delay in bidding queue with bidding function XE (v).

Let v1 satisfy v1W2(v1) = v1W1 −M.

(pE (v),XE (v)) is a stable policy with v1 as above.

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A Theorem

For 0 ≤ t ≤ 1 and a ≤ v1 ≤ b define

pE (v) =

0 for v > v1,

t for v = v1,

1 for v < v1.

F1(v) :=

0 v ≤ v1

F (v)∫ bv1

dF (v)v1 ≤ v ≤ b,

1 v > b.

XE (v) =

∫ v

0

2λ2W0y

(1− ρ2 − ρ2F1(y))3dF1(y)

Let W1 be expected delay in FIFO queue with arrival rate λ1 andW2(v) expected delay in bidding queue with bidding function XE (v).

Let v1 satisfy v1W2(v1) = v1W1 −M.

(pE (v),XE (v)) is a stable policy with v1 as above.

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A Theorem

For 0 ≤ t ≤ 1 and a ≤ v1 ≤ b define

pE (v) =

0 for v > v1,

t for v = v1,

1 for v < v1.

F1(v) :=

0 v ≤ v1

F (v)∫ bv1

dF (v)v1 ≤ v ≤ b,

1 v > b.

XE (v) =

∫ v

0

2λ2W0y

(1− ρ2 − ρ2F1(y))3dF1(y)

Let W1 be expected delay in FIFO queue with arrival rate λ1 andW2(v) expected delay in bidding queue with bidding function XE (v).

Let v1 satisfy v1W2(v1) = v1W1 −M.

(pE (v),XE (v)) is a stable policy with v1 as above.

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A Theorem

For 0 ≤ t ≤ 1 and a ≤ v1 ≤ b define

pE (v) =

0 for v > v1,

t for v = v1,

1 for v < v1.

F1(v) :=

0 v ≤ v1

F (v)∫ bv1

dF (v)v1 ≤ v ≤ b,

1 v > b.

XE (v) =

∫ v

0

2λ2W0y

(1− ρ2 − ρ2F1(y))3dF1(y)

Let W1 be expected delay in FIFO queue with arrival rate λ1 andW2(v) expected delay in bidding queue with bidding function XE (v).

Let v1 satisfy v1W2(v1) = v1W1 −M.

(pE (v),XE (v)) is a stable policy with v1 as above.

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Proof Sketch

For v < v1, we have

M = v1(W1 −W2(v1)) > v(W1 −W2(v1))

This implies vW1 −M < vW2(v1)) and hence for v < v1, we havepE (v) = 1;

For v > v1, using concavity of C (v) and the property thatdC(v)dv = W (x) it can be shown that for ε > 0,

C (v1 + ε) < (v1 + ε)W1 −M

implying that pE (v) = 0.

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Proof Sketch

For v < v1, we have

M = v1(W1 −W2(v1)) > v(W1 −W2(v1))

This implies vW1 −M < vW2(v1)) and hence for v < v1, we havepE (v) = 1;

For v > v1, using concavity of C (v) and the property thatdC(v)dv = W (x) it can be shown that for ε > 0,

C (v1 + ε) < (v1 + ε)W1 −M

implying that pE (v) = 0.

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Discussion

v1 satisfying v1W2(v1) = v1W1 −M is unique.

If p(v) is such that the resulting F (v) has a density, then Wardropequilibrium routing function is of the threshold type

Remark: By moving M to the ‘other side’ it can also be interpretedas the minimum bid in the bidding queue and X (v) will be the‘excess’ bid over M.

v1W2(v1) = v1W1 −M

v1W2(v1) + M = v1W1

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Discussion

v1 satisfying v1W2(v1) = v1W1 −M is unique.

If p(v) is such that the resulting F (v) has a density, then Wardropequilibrium routing function is of the threshold type

Remark: By moving M to the ‘other side’ it can also be interpretedas the minimum bid in the bidding queue and X (v) will be the‘excess’ bid over M.

v1W2(v1) = v1W1 −M

v1W2(v1) + M = v1W1

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Discussion

v1 satisfying v1W2(v1) = v1W1 −M is unique.

If p(v) is such that the resulting F (v) has a density, then Wardropequilibrium routing function is of the threshold type

Remark: By moving M to the ‘other side’ it can also be interpretedas the minimum bid in the bidding queue and X (v) will be the‘excess’ bid over M.

v1W2(v1) = v1W1 −M

v1W2(v1) + M = v1W1

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Discussion (contd.)

For a fixed λ, F (v) and µ2, the revenue rate at equilibrium decreaseswith increasing µ1.

Decreasing λ decreases revenue rate.

Recall that the left truncation of the sensitivity function increases therevenue rate in a single server queue.

This leads us to ask the following question.

Assume that a total capacity of µ is availableOf this capacity, µ1 is allocated to the FIFO queue and µ2 = µ− µ1 isallocated to the bidding queue.How does revenue vary as a function of µ1?

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Discussion (contd.)

For a fixed λ, F (v) and µ2, the revenue rate at equilibrium decreaseswith increasing µ1.

Decreasing λ decreases revenue rate.

Recall that the left truncation of the sensitivity function increases therevenue rate in a single server queue.

This leads us to ask the following question.

Assume that a total capacity of µ is availableOf this capacity, µ1 is allocated to the FIFO queue and µ2 = µ− µ1 isallocated to the bidding queue.How does revenue vary as a function of µ1?

D. Manjunath (IIT Bombay) To jump the queue or wait my turn? January 14, 2014 15 / 24

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Discussion (contd.)

For a fixed λ, F (v) and µ2, the revenue rate at equilibrium decreaseswith increasing µ1.

Decreasing λ decreases revenue rate.

Recall that the left truncation of the sensitivity function increases therevenue rate in a single server queue.

This leads us to ask the following question.

Assume that a total capacity of µ is availableOf this capacity, µ1 is allocated to the FIFO queue and µ2 = µ− µ1 isallocated to the bidding queue.How does revenue vary as a function of µ1?

D. Manjunath (IIT Bombay) To jump the queue or wait my turn? January 14, 2014 15 / 24

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Discussion (contd.)

For a fixed λ, F (v) and µ2, the revenue rate at equilibrium decreaseswith increasing µ1.

Decreasing λ decreases revenue rate.

Recall that the left truncation of the sensitivity function increases therevenue rate in a single server queue.

This leads us to ask the following question.

Assume that a total capacity of µ is availableOf this capacity, µ1 is allocated to the FIFO queue and µ2 = µ− µ1 isallocated to the bidding queue.How does revenue vary as a function of µ1?

D. Manjunath (IIT Bombay) To jump the queue or wait my turn? January 14, 2014 15 / 24

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Discussion (contd.)

For a fixed λ, F (v) and µ2, the revenue rate at equilibrium decreaseswith increasing µ1.

Decreasing λ decreases revenue rate.

Recall that the left truncation of the sensitivity function increases therevenue rate in a single server queue.

This leads us to ask the following question.

Assume that a total capacity of µ is available

Of this capacity, µ1 is allocated to the FIFO queue and µ2 = µ− µ1 isallocated to the bidding queue.How does revenue vary as a function of µ1?

D. Manjunath (IIT Bombay) To jump the queue or wait my turn? January 14, 2014 15 / 24

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Discussion (contd.)

For a fixed λ, F (v) and µ2, the revenue rate at equilibrium decreaseswith increasing µ1.

Decreasing λ decreases revenue rate.

Recall that the left truncation of the sensitivity function increases therevenue rate in a single server queue.

This leads us to ask the following question.

Assume that a total capacity of µ is availableOf this capacity, µ1 is allocated to the FIFO queue and µ2 = µ− µ1 isallocated to the bidding queue.

How does revenue vary as a function of µ1?

D. Manjunath (IIT Bombay) To jump the queue or wait my turn? January 14, 2014 15 / 24

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Discussion (contd.)

For a fixed λ, F (v) and µ2, the revenue rate at equilibrium decreaseswith increasing µ1.

Decreasing λ decreases revenue rate.

Recall that the left truncation of the sensitivity function increases therevenue rate in a single server queue.

This leads us to ask the following question.

Assume that a total capacity of µ is availableOf this capacity, µ1 is allocated to the FIFO queue and µ2 = µ− µ1 isallocated to the bidding queue.How does revenue vary as a function of µ1?

D. Manjunath (IIT Bombay) To jump the queue or wait my turn? January 14, 2014 15 / 24

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Numerical Results

Example 1: λ = 0.5, µ = 1 F (v) = v100 for 0 ≤ v ≤ 100.

Example 2: λ = 0.9, µ = 1 F (v) = v0.5

10 for 0 ≤ v ≤ 100.

Revenue as a function of µ2 with M = 0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

20

40

60

80

100

120

140

160

µ2

Re

ve

nu

e

Example 10 0.2 0.4 0.6 0.8 1

0

500

1000

1500

2000

2500

µ2

Reven

ue

Example 2

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Numerical Results

Example 1: λ = 0.5, µ = 1 F (v) = v100 for 0 ≤ v ≤ 100.

Example 2: λ = 0.9, µ = 1 F (v) = v0.5

10 for 0 ≤ v ≤ 100.

Revenue as a function of µ2 with M = 0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

20

40

60

80

100

120

140

160

µ2

Re

ve

nu

e

Example 1

0 0.2 0.4 0.6 0.8 10

500

1000

1500

2000

2500

µ2

Reven

ue

Example 2

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Numerical Results

Example 1: λ = 0.5, µ = 1 F (v) = v100 for 0 ≤ v ≤ 100.

Example 2: λ = 0.9, µ = 1 F (v) = v0.5

10 for 0 ≤ v ≤ 100.

Revenue as a function of µ2 with M = 0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

20

40

60

80

100

120

140

160

µ2

Re

ve

nu

e

Example 10 0.2 0.4 0.6 0.8 1

0

500

1000

1500

2000

2500

µ2

Re

ve

nu

e

Example 2

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Numerical Results (contd.)

Example 1: λ = 0.5, µ = 1 F (v) = v100 for 0 ≤ v ≤ 100.

Example 2: λ = 0.9, µ = 1 F (v) = v0.5

10 for 0 ≤ v ≤ 100.

Threshold as a function of M for µ1 = 0.5.

0 1000 2000 3000 4000 500060

65

70

75

80

85

90

95

100

M

v1

Example 1

0 2 4 6 8 10

x 104

29

29.5

30

30.5

31

M

v1

Example 2

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Numerical Results (contd.)

Example 1: λ = 0.5, µ = 1 F (v) = v100 for 0 ≤ v ≤ 100.

Example 2: λ = 0.9, µ = 1 F (v) = v0.5

10 for 0 ≤ v ≤ 100.

Threshold as a function of M for µ1 = 0.5.

0 1000 2000 3000 4000 500060

65

70

75

80

85

90

95

100

M

v1

Example 1

0 2 4 6 8 10

x 104

29

29.5

30

30.5

31

M

v1

Example 2

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Numerical Results (contd.)

Example 1: λ = 0.5, µ = 1 F (v) = v100 for 0 ≤ v ≤ 100.

Example 2: λ = 0.9, µ = 1 F (v) = v0.5

10 for 0 ≤ v ≤ 100.

Threshold as a function of M for µ1 = 0.5.

0 1000 2000 3000 4000 500060

65

70

75

80

85

90

95

100

M

v1

Example 1

0 2 4 6 8 10

x 104

29

29.5

30

30.5

31

M

v1

Example 2

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Numerical Results (contd.)Example 1: λ = 0.5, µ = 1 F (v) = v

100 for 0 ≤ v ≤ 100.

Example 2: λ = 0.9, µ = 1 F (v) = v0.5

10 for 0 ≤ v ≤ 100.Let µ2,min be the minimum service rate required to earn positive revenue.

µ2,min as a function of M.

0 500 1000 1500 20000.25

0.3

0.35

0.4

0.45

0.5

M

µ 2 Min

imum

Example 1

0 2000 4000 6000 8000 10000

0.06

0.065

0.07

0.075

0.08

0.085

0.09

0.095

0.1

M

µ 2 min

imum

Example 2

D. Manjunath (IIT Bombay) To jump the queue or wait my turn? January 14, 2014 18 / 24

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Numerical Results (contd.)Example 1: λ = 0.5, µ = 1 F (v) = v

100 for 0 ≤ v ≤ 100.

Example 2: λ = 0.9, µ = 1 F (v) = v0.5

10 for 0 ≤ v ≤ 100.Let µ2,min be the minimum service rate required to earn positive revenue.

µ2,min as a function of M.

0 500 1000 1500 20000.25

0.3

0.35

0.4

0.45

0.5

M

µ 2 Min

imum

Example 1

0 2000 4000 6000 8000 10000

0.06

0.065

0.07

0.075

0.08

0.085

0.09

0.095

0.1

M

µ 2 min

imum

Example 2

D. Manjunath (IIT Bombay) To jump the queue or wait my turn? January 14, 2014 18 / 24

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Numerical Results (contd.)Example 1: λ = 0.5, µ = 1 F (v) = v

100 for 0 ≤ v ≤ 100.

Example 2: λ = 0.9, µ = 1 F (v) = v0.5

10 for 0 ≤ v ≤ 100.Let µ2,min be the minimum service rate required to earn positive revenue.

µ2,min as a function of M.

0 500 1000 1500 20000.25

0.3

0.35

0.4

0.45

0.5

M

µ 2 Min

imum

Example 1

0 2000 4000 6000 8000 10000

0.06

0.065

0.07

0.075

0.08

0.085

0.09

0.095

0.1

Mµ 2 m

inim

um

Example 2

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Lessons

For a fixed µ, the revenue can actually increase if some capacity isallocated to a FIFO queue.

If µ1 < λ then some arrivals will choose the bidding queue and therewill be non zero revenue.

However, if µ1 > λ, then for sufficiently large M, there can be norevenue.

Now interpret the server as a babu (public service official(?)), the bidsas bribes, M as society’s reward to the customers for choosing therighteous way.

There is more money to be made by being partially honest than beingfully dishonest.

Of course, the latter conclusions are in jest and not meant to be usedin a sociological study.

D. Manjunath (IIT Bombay) To jump the queue or wait my turn? January 14, 2014 19 / 24

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Lessons

For a fixed µ, the revenue can actually increase if some capacity isallocated to a FIFO queue.

If µ1 < λ then some arrivals will choose the bidding queue and therewill be non zero revenue.

However, if µ1 > λ, then for sufficiently large M, there can be norevenue.

Now interpret the server as a babu (public service official(?)), the bidsas bribes, M as society’s reward to the customers for choosing therighteous way.

There is more money to be made by being partially honest than beingfully dishonest.

Of course, the latter conclusions are in jest and not meant to be usedin a sociological study.

D. Manjunath (IIT Bombay) To jump the queue or wait my turn? January 14, 2014 19 / 24

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Lessons

For a fixed µ, the revenue can actually increase if some capacity isallocated to a FIFO queue.

If µ1 < λ then some arrivals will choose the bidding queue and therewill be non zero revenue.

However, if µ1 > λ, then for sufficiently large M, there can be norevenue.

Now interpret the server as a babu (public service official(?)), the bidsas bribes, M as society’s reward to the customers for choosing therighteous way.

There is more money to be made by being partially honest than beingfully dishonest.

Of course, the latter conclusions are in jest and not meant to be usedin a sociological study.

D. Manjunath (IIT Bombay) To jump the queue or wait my turn? January 14, 2014 19 / 24

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Lessons

For a fixed µ, the revenue can actually increase if some capacity isallocated to a FIFO queue.

If µ1 < λ then some arrivals will choose the bidding queue and therewill be non zero revenue.

However, if µ1 > λ, then for sufficiently large M, there can be norevenue.

Now interpret the server as a babu (public service official(?)), the bidsas bribes, M as society’s reward to the customers for choosing therighteous way.

There is more money to be made by being partially honest than beingfully dishonest.

Of course, the latter conclusions are in jest and not meant to be usedin a sociological study.

D. Manjunath (IIT Bombay) To jump the queue or wait my turn? January 14, 2014 19 / 24

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Lessons

For a fixed µ, the revenue can actually increase if some capacity isallocated to a FIFO queue.

If µ1 < λ then some arrivals will choose the bidding queue and therewill be non zero revenue.

However, if µ1 > λ, then for sufficiently large M, there can be norevenue.

Now interpret the server as a babu (public service official(?)), the bidsas bribes, M as society’s reward to the customers for choosing therighteous way.

There is more money to be made by being partially honest than beingfully dishonest.

Of course, the latter conclusions are in jest and not meant to be usedin a sociological study.

D. Manjunath (IIT Bombay) To jump the queue or wait my turn? January 14, 2014 19 / 24

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Lessons

For a fixed µ, the revenue can actually increase if some capacity isallocated to a FIFO queue.

If µ1 < λ then some arrivals will choose the bidding queue and therewill be non zero revenue.

However, if µ1 > λ, then for sufficiently large M, there can be norevenue.

Now interpret the server as a babu (public service official(?)), the bidsas bribes, M as society’s reward to the customers for choosing therighteous way.

There is more money to be made by being partially honest than beingfully dishonest.

Of course, the latter conclusions are in jest and not meant to be usedin a sociological study.

D. Manjunath (IIT Bombay) To jump the queue or wait my turn? January 14, 2014 19 / 24

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Overtime you say?

Now consider what happens if we add capacity to the FIFO queuewithout affecting the bidding queue.

Specifically, we study the revenue at equilibrium as a function of µ1with µ2 and all other parameters (M, F (v), and λ) being fixed.

Notation is simplified to R(µ1) for the revenue rate as a function ofµ1.

Lemma

R(0) ≥ R(µ1)

i.e., adding capacity in the form of FIFO queue does not increase therevenue.

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Overtime you say?

Now consider what happens if we add capacity to the FIFO queuewithout affecting the bidding queue.

Specifically, we study the revenue at equilibrium as a function of µ1with µ2 and all other parameters (M, F (v), and λ) being fixed.

Notation is simplified to R(µ1) for the revenue rate as a function ofµ1.

Lemma

R(0) ≥ R(µ1)

i.e., adding capacity in the form of FIFO queue does not increase therevenue.

D. Manjunath (IIT Bombay) To jump the queue or wait my turn? January 14, 2014 20 / 24

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Overtime you say?

Now consider what happens if we add capacity to the FIFO queuewithout affecting the bidding queue.

Specifically, we study the revenue at equilibrium as a function of µ1with µ2 and all other parameters (M, F (v), and λ) being fixed.

Notation is simplified to R(µ1) for the revenue rate as a function ofµ1.

Lemma

R(0) ≥ R(µ1)

i.e., adding capacity in the form of FIFO queue does not increase therevenue.

D. Manjunath (IIT Bombay) To jump the queue or wait my turn? January 14, 2014 20 / 24

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Overtime you say?

Now consider what happens if we add capacity to the FIFO queuewithout affecting the bidding queue.

Specifically, we study the revenue at equilibrium as a function of µ1with µ2 and all other parameters (M, F (v), and λ) being fixed.

Notation is simplified to R(µ1) for the revenue rate as a function ofµ1.

Lemma

R(0) ≥ R(µ1)

i.e., adding capacity in the form of FIFO queue does not increase therevenue.

D. Manjunath (IIT Bombay) To jump the queue or wait my turn? January 14, 2014 20 / 24

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Giving Up: The Balking Case

Balking : If the service provided has a value, say P, then thecustomers whose total cost C (v) exceeds P prefer not to take theservice.

In such a balking model P = X (v∗) + v∗W (v∗) where v∗ is thehighest sensitivity of a customer joining the queue and customers withv > v∗ balk.

Let v∗(µ1) be the highest sensitivity of customers joining the biddingqueue when the service rate of the FIFO queue is µ1.

D. Manjunath (IIT Bombay) To jump the queue or wait my turn? January 14, 2014 21 / 24

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Giving Up: The Balking Case

Balking : If the service provided has a value, say P, then thecustomers whose total cost C (v) exceeds P prefer not to take theservice.

In such a balking model P = X (v∗) + v∗W (v∗) where v∗ is thehighest sensitivity of a customer joining the queue and customers withv > v∗ balk.

Let v∗(µ1) be the highest sensitivity of customers joining the biddingqueue when the service rate of the FIFO queue is µ1.

D. Manjunath (IIT Bombay) To jump the queue or wait my turn? January 14, 2014 21 / 24

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Giving Up: The Balking Case

Balking : If the service provided has a value, say P, then thecustomers whose total cost C (v) exceeds P prefer not to take theservice.

In such a balking model P = X (v∗) + v∗W (v∗) where v∗ is thehighest sensitivity of a customer joining the queue and customers withv > v∗ balk.

Let v∗(µ1) be the highest sensitivity of customers joining the biddingqueue when the service rate of the FIFO queue is µ1.

D. Manjunath (IIT Bombay) To jump the queue or wait my turn? January 14, 2014 21 / 24

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Results

Theorem

For uniform sensitivity profile F (v) = Av , and fixed µ2,

v∗(µ1) increases with µ1.

R(µ1) decreases with µ1, . i.e., adding a FIFO queue decreasesrevenue.

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Results

Theorem

For uniform sensitivity profile F (v) = Av , and fixed µ2,

v∗(µ1) increases with µ1.

R(µ1) decreases with µ1, . i.e., adding a FIFO queue decreasesrevenue.

D. Manjunath (IIT Bombay) To jump the queue or wait my turn? January 14, 2014 22 / 24

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Results

Theorem

For uniform sensitivity profile F (v) = Av , and fixed µ2,

v∗(µ1) increases with µ1.

R(µ1) decreases with µ1, . i.e., adding a FIFO queue decreasesrevenue.

D. Manjunath (IIT Bombay) To jump the queue or wait my turn? January 14, 2014 22 / 24

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Concluding Remarks

A motivation

Primary motivation came from developing bidding models for servicesystems with queues and arrivals.

For such a scenario, an obvious variation is to have processor sharingand allocate the weights in proportion to the bids.

A secondary motivation is to explore new models to test Myrdal’sclaim that corruption in public service decreases efficiency.

In this context, M was a simplistic model for reward. Other rewardstructures are being explored.

D. Manjunath (IIT Bombay) To jump the queue or wait my turn? January 14, 2014 23 / 24

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Concluding Remarks

A motivation

Primary motivation came from developing bidding models for servicesystems with queues and arrivals.

For such a scenario, an obvious variation is to have processor sharingand allocate the weights in proportion to the bids.

A secondary motivation is to explore new models to test Myrdal’sclaim that corruption in public service decreases efficiency.

In this context, M was a simplistic model for reward. Other rewardstructures are being explored.

D. Manjunath (IIT Bombay) To jump the queue or wait my turn? January 14, 2014 23 / 24

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Concluding Remarks

A motivation

Primary motivation came from developing bidding models for servicesystems with queues and arrivals.

For such a scenario, an obvious variation is to have processor sharingand allocate the weights in proportion to the bids.

A secondary motivation is to explore new models to test Myrdal’sclaim that corruption in public service decreases efficiency.

In this context, M was a simplistic model for reward. Other rewardstructures are being explored.

D. Manjunath (IIT Bombay) To jump the queue or wait my turn? January 14, 2014 23 / 24

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Concluding Remarks

A motivation

Primary motivation came from developing bidding models for servicesystems with queues and arrivals.

For such a scenario, an obvious variation is to have processor sharingand allocate the weights in proportion to the bids.

A secondary motivation is to explore new models to test Myrdal’sclaim that corruption in public service decreases efficiency.

In this context, M was a simplistic model for reward. Other rewardstructures are being explored.

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Acknowledgements

The first part is a summary from

L. Kleinrock, “Optimal bribing for queue position,” OperationsResearch, vol 15, pp. 304–318, 1967.F. Lui, “An equilibrium model of bribery,” Political Economy, vol. 93,pp. 760–781, 1985.A. Glazer and R. Hassin, “Stable priority purchasing in ueues,”Operations Research Letters, vol. 4, no. 6, pp. 285–288, 1986.

Joint work with Murtuza Ali Abidin (IITB) and Tejas Bodas (IITB).

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Acknowledgements

The first part is a summary from

L. Kleinrock, “Optimal bribing for queue position,” OperationsResearch, vol 15, pp. 304–318, 1967.

F. Lui, “An equilibrium model of bribery,” Political Economy, vol. 93,pp. 760–781, 1985.A. Glazer and R. Hassin, “Stable priority purchasing in ueues,”Operations Research Letters, vol. 4, no. 6, pp. 285–288, 1986.

Joint work with Murtuza Ali Abidin (IITB) and Tejas Bodas (IITB).

D. Manjunath (IIT Bombay) To jump the queue or wait my turn? January 14, 2014 24 / 24

Page 100: To jump the queue or wait my turn? - ERNETifcam/new_avenue/Slides/Seminars/Manjunath.pdfD. Manjunath (IIT Bombay)To jump the queue or wait my turn?January 14, 2014 4 / 24 Adding value

Acknowledgements

The first part is a summary from

L. Kleinrock, “Optimal bribing for queue position,” OperationsResearch, vol 15, pp. 304–318, 1967.F. Lui, “An equilibrium model of bribery,” Political Economy, vol. 93,pp. 760–781, 1985.

A. Glazer and R. Hassin, “Stable priority purchasing in ueues,”Operations Research Letters, vol. 4, no. 6, pp. 285–288, 1986.

Joint work with Murtuza Ali Abidin (IITB) and Tejas Bodas (IITB).

D. Manjunath (IIT Bombay) To jump the queue or wait my turn? January 14, 2014 24 / 24

Page 101: To jump the queue or wait my turn? - ERNETifcam/new_avenue/Slides/Seminars/Manjunath.pdfD. Manjunath (IIT Bombay)To jump the queue or wait my turn?January 14, 2014 4 / 24 Adding value

Acknowledgements

The first part is a summary from

L. Kleinrock, “Optimal bribing for queue position,” OperationsResearch, vol 15, pp. 304–318, 1967.F. Lui, “An equilibrium model of bribery,” Political Economy, vol. 93,pp. 760–781, 1985.A. Glazer and R. Hassin, “Stable priority purchasing in ueues,”Operations Research Letters, vol. 4, no. 6, pp. 285–288, 1986.

Joint work with Murtuza Ali Abidin (IITB) and Tejas Bodas (IITB).

D. Manjunath (IIT Bombay) To jump the queue or wait my turn? January 14, 2014 24 / 24

Page 102: To jump the queue or wait my turn? - ERNETifcam/new_avenue/Slides/Seminars/Manjunath.pdfD. Manjunath (IIT Bombay)To jump the queue or wait my turn?January 14, 2014 4 / 24 Adding value

Acknowledgements

The first part is a summary from

L. Kleinrock, “Optimal bribing for queue position,” OperationsResearch, vol 15, pp. 304–318, 1967.F. Lui, “An equilibrium model of bribery,” Political Economy, vol. 93,pp. 760–781, 1985.A. Glazer and R. Hassin, “Stable priority purchasing in ueues,”Operations Research Letters, vol. 4, no. 6, pp. 285–288, 1986.

Joint work with Murtuza Ali Abidin (IITB) and Tejas Bodas (IITB).

D. Manjunath (IIT Bombay) To jump the queue or wait my turn? January 14, 2014 24 / 24