[PPT] - To jump the queue or wait my turn? D. Manjunath IIT Bombay January PowerPoint Presentation

SLIDE 1

To jump the queue or wait my turn?

D. Manjunath

IIT Bombay

January 14, 2014

D. Manjunath (IIT Bombay)

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SLIDE 2

Bidding for priorities

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

D. Manjunath (IIT Bombay)

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SLIDE 3

Bidding for priorities

A single server queue has customers arriving according to a homogeneous Poisson process of rate λ. Service times are i.i.d. with distribution B(y) and mean 1

µ; Define

ρ := λ

µ.

D. Manjunath (IIT Bombay)

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SLIDE 4

Bidding for priorities

A single server queue has customers arriving according to a homogeneous Poisson process of rate λ. Service times are i.i.d. with distribution B(y) and mean 1

µ; Define

ρ := λ

µ.

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

D. Manjunath (IIT Bombay)

To jump the queue or wait my turn? January 14, 2014 2 / 24

SLIDE 5

Bidding for priorities

A single server queue has customers arriving according to a homogeneous Poisson process of rate λ. Service times are i.i.d. with distribution B(y) and mean 1

µ; Define

ρ := λ

µ.

There is no balking from the system; all arrivals receive service. An arriving customer bids to pay X rupees for service.

D. Manjunath (IIT Bombay)

To jump the queue or wait my turn? January 14, 2014 2 / 24

SLIDE 6

Bidding for priorities

A single server queue has customers arriving according to a homogeneous Poisson process of rate λ. Service times are i.i.d. with distribution B(y) and mean 1

µ; Define

ρ := λ

µ.

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 or non preemptive priority service.

D. Manjunath (IIT Bombay)

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SLIDE 7

Bidding for priorities

A single server queue has customers arriving according to a homogeneous Poisson process of rate λ. Service times are i.i.d. with distribution B(y) and mean 1

µ; Define

ρ := λ

µ.

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 or non preemptive priority service. Assume that the bids are i.i.d. and chosen according to a continuous distribution B(x).

D. Manjunath (IIT Bombay)

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SLIDE 8

Bidding for priorities

Sojourn time of customer bidding x, W (x), has three components.

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SLIDE 9

Bidding for priorities

Sojourn time of customer bidding x, W (x), has three components.

Its own service time which has mean 1

µ.

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SLIDE 10

Bidding for priorities

Sojourn time of customer bidding x, W (x), has three components.

Its own service time which has mean 1

µ.

Waiting caused by those already in system and having bids higher than

x. From Little’s law and PASTA property, the expectation will be

1 µ ∞

x

W (y) (λdB(y))

.
D. Manjunath (IIT Bombay)

To jump the queue or wait my turn? January 14, 2014 3 / 24

SLIDE 11

Bidding for priorities

Sojourn time of customer bidding x, W (x), has three components.

Its own service time which has mean 1

µ.

Waiting caused by those already in system and having bids higher than

x. 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 stay in the system and bid higher than x. This is 1 µ W (x) ∞

x

λdB(y)

.
D. Manjunath (IIT Bombay)

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SLIDE 12

Bidding for priorities

Sojourn time of customer bidding x, W (x), has three components.

Its own service time which has mean 1

µ.

Waiting caused by those already in system and having bids higher than

x. 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 stay in the system and bid higher than x. This is 1 µ W (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 non preemptive priority service and 1/µ for preemptive priority service.

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SLIDE 13

Adding value to time

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

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SLIDE 14

Adding value to time

Now let each customer have a non negative delay sensitivity parameter V . Specifically, let V be cost per unit waiting time. Assume that V of an arriving customer is i.i.d. with distribution F(v), F(v) has a density with support in (a, b), i.e., F(v) =

0 < a < v

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

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SLIDE 15

Adding value to time

Now let each customer have a non negative delay sensitivity parameter V . Specifically, let V be cost per unit waiting time. Assume that V of an arriving customer is i.i.d. with distribution F(v), F(v) has a density with support in (a, b), i.e., F(v) =

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.

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SLIDE 16

Adding value to time

Now let each customer have a non negative delay sensitivity parameter V . Specifically, let V be cost per unit waiting time. Assume that V of an arriving customer is i.i.d. with distribution F(v), F(v) has a density with support in (a, b), i.e., F(v) =

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)).

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SLIDE 17

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 under policy X.

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SLIDE 18

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 under policy X. Now assume customers are strategic—X(v) minimises total cost C(v).

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SLIDE 19

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 under policy X. Now assume customers are strategic—X(v) minimises total cost C(v). Bidding function X(v) is said to be in equilibrium if no individual customer has an incentive to deviate from it.

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SLIDE 20

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 under policy X. Now assume customers are strategic—X(v) minimises total cost C(v). Bidding function X(v) is said to be in equilibrium if no individual customer has an incentive to deviate from it. If customers follow an equilibrium bidding policy, then C(v) is minimum for each v, and a customer of sensitivity v cannot unilaterally deviate from the equilibrium X(v) and lower its C(v).

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SLIDE 21

Characterising stable policies

Lemma

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

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SLIDE 22

Characterising stable policies

Lemma

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

Property

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

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SLIDE 23

Characterising stable policies

Lemma

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

Property

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

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

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SLIDE 24

Characterising stable policies

Lemma

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

Property

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

1 W X(x) is strictly decreasing in x. 2 C X(v) is continuous and strictly increasing. Further, it is concave in

v and dC(v)

dv

= W (x).

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SLIDE 25

Characterising stable policies

Lemma

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

Property

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

1 W X(x) is strictly decreasing in x. 2 C X(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).

D. Manjunath (IIT Bombay)

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SLIDE 26

Characterising stable policies

Lemma

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

Property

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

1 W X(x) is strictly decreasing in x. 2 C X(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).

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SLIDE 27

Revenue to the server

For v ∈ (a, b), X(v) = v 2λW0y (1 − ρ − ρF(y))3 dF(y) is a stable bidding policy. W0 is as before—mean residual service time for non preemptive priority policy and 1/mu for preemptive prioirity policy.

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SLIDE 28

Revenue to the server

For v ∈ (a, b), X(v) = v 2λW0y (1 − ρ − ρF(y))3 dF(y) is a stable bidding policy. W0 is as before—mean residual service time for non preemptive priority policy and 1/mu for preemptive prioirity policy. Fixing the arrival rate and the delay sensitivity function F(v), a bidding policy X(v) will yield an expected revenue rate of RX(λ, F(v)) = λ

v

X(v)dF(v).

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SLIDE 29

Properties of the Revenue Function

1 Let F1(v) be a sensitivity profile and F2(v) = F1(v − v0) where

v0 > 0. Then R(λ, F1(v)) < R(λ, F2(v)).

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SLIDE 30

Properties of the Revenue Function

1 Let F1(v) be a sensitivity profile and F2(v) = F1(v − v0) where

v0 > 0. Then R(λ, F1(v)) < R(λ, F2(v)).

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

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SLIDE 31

Properties of the Revenue Function

1 Let F1(v) be a sensitivity profile and F2(v) = F1(v − v0) where

v0 > 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) for

α > 0. Then R(λ, F(v)) < R(λ, F(αv)) when α > 1.

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SLIDE 32

Properties of the Revenue Function

1 Let F1(v) be a sensitivity profile and F2(v) = F1(v − v0) where

v0 > 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) for

α > 0. Then R(λ, F(v)) < R(λ, F(αv)) when α > 1.

4 Let a < a1 < b, and define

F1(v) =    v < a1

F(v) b

a1 dF(x)

v ≥ a1 Then R(λ, F(v)) < R(λ, F1(v)).

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SLIDE 33

Introducing a Second Queue

To the bidding queue, add a parallel FIFO queue that has service rate µ1 while the bidding queue has service rate µ2.

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SLIDE 34

Introducing a Second Queue

To the bidding queue, add a parallel FIFO queue that has service rate µ1 while the bidding queue has service rate µ2. Thus arriving customers make two decisions—(1) which queue to join, and (2) if joining the bidding queue, what should be the bid.

D. Manjunath (IIT Bombay)

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SLIDE 35

Introducing a Second Queue

To the bidding queue, add a parallel FIFO queue that has service rate µ1 while the bidding queue has service rate µ2. Thus arriving customers make two decisions—(1) which queue to join, and (2) if joining the bidding queue, what should be the bid. There is no balking.

D. Manjunath (IIT Bombay)

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SLIDE 36

Introducing a Second Queue

To the bidding queue, add a parallel FIFO queue that has service rate µ1 while the bidding queue has service rate µ2. Thus arriving customers make two decisions—(1) which queue to join, 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, he receives a reward of M rupees.

D. Manjunath (IIT Bombay)

To jump the queue or wait my turn? January 14, 2014 9 / 24

SLIDE 37

Introducing a Second Queue

To the bidding queue, add a parallel FIFO queue that has service rate µ1 while the bidding queue has service rate µ2. Thus arriving customers make two decisions—(1) which queue to join, 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, he receives a reward of M rupees. Like before, we assume oblivious routing and oblivious bidding.

D. Manjunath (IIT Bombay)

To jump the queue or wait my turn? January 14, 2014 9 / 24

SLIDE 38

Introducing a Second Queue

To the bidding queue, add a parallel FIFO queue that has service rate µ1 while the bidding queue has service rate µ2. Thus arriving customers make two decisions—(1) which queue to join, 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, he receives a reward of M rupees. Like before, we assume oblivious routing and oblivious bidding. Customers are strategic—each one minimises its individual cost function.

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SLIDE 39

More Notation

An arrival with sensitivity v has the following decisions

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SLIDE 40

More Notation

An arrival with sensitivity v has the following decisions

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

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SLIDE 41

More Notation

An arrival with sensitivity v has the following decisions

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

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SLIDE 42

More Notation

An arrival with sensitivity v has the following decisions

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

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SLIDE 43

More Notation

An arrival with sensitivity v has the following decisions

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

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SLIDE 44

More Notation

An arrival with sensitivity v has the following decisions

Which queue? Chooses FIFO queue with probability p(v). p(v) : ℜ+ → [0, 1] is called the routing function. What price? Bid X(v) to be paid if it joins bidding queue; X(v) : ℜ+ → ℜ+ 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 queue ρ1 = λ1/µ1 and ρ2 = λ2/µ2.

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SLIDE 45

Remarks

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

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SLIDE 46

Remarks

Delay in FIFO queue is independent of the class; W1 is the expectation. Delay in bidding queue is a function of v; W2(v) is the expectation.

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SLIDE 47

Remarks

Delay in FIFO queue is independent of the class; W1 is the expectation. Delay in bidding queue is a function of v; W2(v) is the expectation. Let X E(v) be a equilibrium strategy; equilibrium will be Wardrop and the following will be true.

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SLIDE 48

Remarks

Delay in FIFO queue is independent of the class; W1 is the expectation. Delay in bidding queue is a function of v; W2(v) is the expectation. Let X E(v) be a equilibrium strategy; equilibrium will be Wardrop and the following will be true.

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

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SLIDE 49

Remarks

Delay in FIFO queue is independent of the class; W1 is the expectation. Delay in bidding queue is a function of v; W2(v) is the expectation. Let X E(v) be a equilibrium strategy; equilibrium will be Wardrop and the following will be true.

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

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SLIDE 50

Remarks

Delay in FIFO queue is independent of the class; W1 is the expectation. Delay in bidding queue is a function of v; W2(v) is the expectation. Let X E(v) be a equilibrium strategy; equilibrium will be Wardrop and the following will be true.

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

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SLIDE 51

A Theorem

For 0 ≤ t ≤ 1 and a ≤ v1 ≤ b define X E(v) = v 2λ2W0y (1 − ρ2 − ρ2F1(y))3 dF1(y) (pE(v), X E(v)) is a stable policy with v1 as above.

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SLIDE 52

A Theorem

For 0 ≤ t ≤ 1 and a ≤ v1 ≤ b define X E(v) = v 2λ2W0y (1 − ρ2 − ρ2F1(y))3 dF1(y) (pE(v), X E(v)) is a stable policy with v1 as above.

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SLIDE 53

A Theorem

For 0 ≤ t ≤ 1 and a ≤ v1 ≤ b define pE(v) =      for v > v1, t for v = v1, 1 for v < v1. X E(v) = v 2λ2W0y (1 − ρ2 − ρ2F1(y))3 dF1(y) (pE(v), X E(v)) is a stable policy with v1 as above.

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SLIDE 54

A Theorem

For 0 ≤ t ≤ 1 and a ≤ v1 ≤ b define pE(v) =      for v > v1, t for v = v1, 1 for v < v1. F1(v) :=        v ≤ v1

F(v) b

v1 dF(v)

v1 ≤ v ≤ b, 1 v > b. X E(v) = v 2λ2W0y (1 − ρ2 − ρ2F1(y))3 dF1(y) (pE(v), X E(v)) is a stable policy with v1 as above.

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SLIDE 55

A Theorem

For 0 ≤ t ≤ 1 and a ≤ v1 ≤ b define pE(v) =      for v > v1, t for v = v1, 1 for v < v1. F1(v) :=        v ≤ v1

F(v) b

v1 dF(v)

v1 ≤ v ≤ b, 1 v > b. X E(v) = v 2λ2W0y (1 − ρ2 − ρ2F1(y))3 dF1(y) Let W1 be expected delay in FIFO queue with arrival rate λ1 and W2(v) expected delay in bidding queue with bidding function X E(v). (pE(v), X E(v)) is a stable policy with v1 as above.

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SLIDE 56

A Theorem

For 0 ≤ t ≤ 1 and a ≤ v1 ≤ b define pE(v) =      for v > v1, t for v = v1, 1 for v < v1. F1(v) :=        v ≤ v1

F(v) b

v1 dF(v)

v1 ≤ v ≤ b, 1 v > b. X E(v) = v 2λ2W0y (1 − ρ2 − ρ2F1(y))3 dF1(y) Let W1 be expected delay in FIFO queue with arrival rate λ1 and W2(v) expected delay in bidding queue with bidding function X E(v). Let v1 satisfy v1W2(v1) = v1W1 − M. (pE(v), X E(v)) is a stable policy with v1 as above.

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SLIDE 57

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 have pE(v) = 1;

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SLIDE 58

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 have pE(v) = 1; For v > v1, using concavity of C(v) and the property that

dC(v) dv

= W (x) it can be shown that for ǫ > 0, C(v1 + ǫ) < (v1 + ǫ)W1 − M implying that pE(v) = 0.

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SLIDE 59

Discussion

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

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SLIDE 60

Discussion

v1 satisfying v1W2(v1) = v1W1 − M is unique. If p(v) is such that the resulting F(v) has a density, then Wardrop equilibrium routing function is of the threshold type

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SLIDE 61

Discussion

v1 satisfying v1W2(v1) = v1W1 − M is unique. If p(v) is such that the resulting F(v) has a density, then Wardrop equilibrium routing function is of the threshold type Remark: By moving M to the ‘other side’ it can also be interpreted as 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

D. Manjunath (IIT Bombay)

To jump the queue or wait my turn? January 14, 2014 14 / 24

SLIDE 62