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

Characterising stable policies Lemma If X ( v ) is a stable bidding function, B X ( 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 ) = W ( x ) . dv D. Manjunath (IIT Bombay) To jump the queue or wait my turn? January 14, 2014 6 / 24

Characterising stable policies Lemma If X ( v ) is a stable bidding function, B X ( 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 ) = W ( x ) . dv 3 If v 1 < v 2 then X ( v 1 ) < X ( v 2 ) . D. Manjunath (IIT Bombay) To jump the queue or wait my turn? January 14, 2014 6 / 24

Characterising stable policies Lemma If X ( v ) is a stable bidding function, B X ( 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 ) = W ( x ) . dv 3 If v 1 < v 2 then X ( v 1 ) < X ( v 2 ) . 4 B ( X ( v )) = F ( v ) . D. Manjunath (IIT Bombay) To jump the queue or wait my turn? January 14, 2014 6 / 24

Revenue to the server For v ∈ ( a , b ) , � v 2 λ W 0 y X ( v ) = (1 − ρ − ρ F ( y )) 3 dF ( y ) 0 is a stable bidding policy. W 0 is as before—mean residual service time for non preemptive priority policy and 1 / mu for preemptive prioirity policy. D. Manjunath (IIT Bombay) To jump the queue or wait my turn? January 14, 2014 7 / 24

Revenue to the server For v ∈ ( a , b ) , � v 2 λ W 0 y X ( v ) = (1 − ρ − ρ F ( y )) 3 dF ( y ) 0 is a stable bidding policy. W 0 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 � R X ( λ, F ( v )) = λ X ( v ) dF ( v ) . v D. Manjunath (IIT Bombay) To jump the queue or wait my turn? January 14, 2014 7 / 24

Properties of the Revenue Function 1 Let F 1 ( v ) be a sensitivity profile and F 2 ( v ) = F 1 ( v − v 0 ) where v 0 > 0 . Then R ( λ, F 1 ( v )) < R ( λ, F 2 ( v )). D. Manjunath (IIT Bombay) To jump the queue or wait my turn? January 14, 2014 8 / 24

Properties of the Revenue Function 1 Let F 1 ( v ) be a sensitivity profile and F 2 ( v ) = F 1 ( v − v 0 ) where v 0 > 0 . Then R ( λ, F 1 ( v )) < R ( λ, F 2 ( v )). 2 If λ 1 < λ 2 then R ( λ 1 , F ( v )) < R ( λ 2 , F ( v )) D. Manjunath (IIT Bombay) To jump the queue or wait my turn? January 14, 2014 8 / 24

Properties of the Revenue Function 1 Let F 1 ( v ) be a sensitivity profile and F 2 ( v ) = F 1 ( v − v 0 ) where v 0 > 0 . Then R ( λ, F 1 ( v )) < R ( λ, F 2 ( v )). 2 If λ 1 < λ 2 then R ( λ 1 , F ( v )) < R ( λ 2 , F ( v )) 3 Consider the scaling of the sensitivity profile, i.e., consider F ( α v ) for α > 0 . Then R ( λ, F ( v )) < R ( λ, F ( α v )) when α > 1 . D. Manjunath (IIT Bombay) To jump the queue or wait my turn? January 14, 2014 8 / 24

Properties of the Revenue Function 1 Let F 1 ( v ) be a sensitivity profile and F 2 ( v ) = F 1 ( v − v 0 ) where v 0 > 0 . Then R ( λ, F 1 ( v )) < R ( λ, F 2 ( v )). 2 If λ 1 < λ 2 then R ( λ 1 , F ( v )) < R ( λ 2 , F ( v )) 3 Consider the scaling of the sensitivity profile, i.e., consider F ( α v ) for α > 0 . Then R ( λ, F ( v )) < R ( λ, F ( α v )) when α > 1 . 4 Let a < a 1 < b , and define  0 v < a 1  F 1 ( v ) = F ( v ) v ≥ a 1 � b a 1 dF ( x )  Then R ( λ, F ( v )) < R ( λ, F 1 ( v )) . D. Manjunath (IIT Bombay) To jump the queue or wait my turn? January 14, 2014 8 / 24

Introducing a Second Queue To the bidding queue, add a parallel FIFO queue that has service rate µ 1 while the bidding queue has service rate µ 2 . D. Manjunath (IIT Bombay) To jump the queue or wait my turn? January 14, 2014 9 / 24

Introducing a Second Queue To the bidding queue, add a parallel FIFO queue that has service rate µ 1 while the bidding queue has service rate µ 2 . 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) To jump the queue or wait my turn? January 14, 2014 9 / 24

Introducing a Second Queue To the bidding queue, add a parallel FIFO queue that has service rate µ 1 while the bidding queue has service rate µ 2 . 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) To jump the queue or wait my turn? January 14, 2014 9 / 24

Introducing a Second Queue To the bidding queue, add a parallel FIFO queue that has service rate µ 1 while the bidding queue has service rate µ 2 . 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

Introducing a Second Queue To the bidding queue, add a parallel FIFO queue that has service rate µ 1 while the bidding queue has service rate µ 2 . 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

Introducing a Second Queue To the bidding queue, add a parallel FIFO queue that has service rate µ 1 while the bidding queue has service rate µ 2 . 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. D. Manjunath (IIT Bombay) To jump the queue or wait my turn? January 14, 2014 9 / 24

More Notation An arrival with sensitivity v has the following decisions D. Manjunath (IIT Bombay) To jump the queue or wait my turn? January 14, 2014 10 / 24

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. D. Manjunath (IIT Bombay) To jump the queue or wait my turn? January 14, 2014 10 / 24

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. D. Manjunath (IIT Bombay) To jump the queue or wait my turn? January 14, 2014 10 / 24

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 . D. Manjunath (IIT Bombay) To jump the queue or wait my turn? January 14, 2014 10 / 24

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. D. Manjunath (IIT Bombay) To jump the queue or wait my turn? January 14, 2014 10 / 24

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. � ∞ λ 1 = λ 0 p ( v ) dF ( v ) is the arrival rate to the FIFO queue λ 2 = λ − λ 1 is the arrival rate to the bidding queue ρ 1 = λ 1 /µ 1 and ρ 2 = λ 2 /µ 2 . D. Manjunath (IIT Bombay) To jump the queue or wait my turn? January 14, 2014 10 / 24

Remarks Delay in FIFO queue is independent of the class; W 1 is the expectation. D. Manjunath (IIT Bombay) To jump the queue or wait my turn? January 14, 2014 11 / 24

Remarks Delay in FIFO queue is independent of the class; W 1 is the expectation. Delay in bidding queue is a function of v ; W 2 ( v ) is the expectation. D. Manjunath (IIT Bombay) To jump the queue or wait my turn? January 14, 2014 11 / 24

Remarks Delay in FIFO queue is independent of the class; W 1 is the expectation. Delay in bidding queue is a function of v ; W 2 ( v ) is the expectation. Let X E ( v ) be a equilibrium strategy; equilibrium will be Wardrop and the following will be true. D. Manjunath (IIT Bombay) To jump the queue or wait my turn? January 14, 2014 11 / 24

Remarks Delay in FIFO queue is independent of the class; W 1 is the expectation. Delay in bidding queue is a function of v ; W 2 ( 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 vW 1 < X ( v ) + W 2 ( X E ( v )) . D. Manjunath (IIT Bombay) To jump the queue or wait my turn? January 14, 2014 11 / 24

Remarks Delay in FIFO queue is independent of the class; W 1 is the expectation. Delay in bidding queue is a function of v ; W 2 ( 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 vW 1 < X ( v ) + W 2 ( X E ( v )) . p ( v ) = 0 implies vW 1 > X ( v ) + W 2 ( X E ( v )) . D. Manjunath (IIT Bombay) To jump the queue or wait my turn? January 14, 2014 11 / 24

Remarks Delay in FIFO queue is independent of the class; W 1 is the expectation. Delay in bidding queue is a function of v ; W 2 ( 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 vW 1 < X ( v ) + W 2 ( X E ( v )) . p ( v ) = 0 implies vW 1 > X ( v ) + W 2 ( X E ( v )) . 0 < p ( v ) < 1 implies vW 1 − M = X E ( v ) + W 2 ( X E ( v )) . D. Manjunath (IIT Bombay) To jump the queue or wait my turn? January 14, 2014 11 / 24

A Theorem For 0 ≤ t ≤ 1 and a ≤ v 1 ≤ b define � v 2 λ 2 W 0 y X E ( v ) = (1 − ρ 2 − ρ 2 F 1 ( y )) 3 dF 1 ( y ) 0 ( p E ( v ) , X E ( v )) is a stable policy with v 1 as above. D. Manjunath (IIT Bombay) To jump the queue or wait my turn? January 14, 2014 12 / 24

A Theorem For 0 ≤ t ≤ 1 and a ≤ v 1 ≤ b define � v 2 λ 2 W 0 y X E ( v ) = (1 − ρ 2 − ρ 2 F 1 ( y )) 3 dF 1 ( y ) 0 ( p E ( v ) , X E ( v )) is a stable policy with v 1 as above. D. Manjunath (IIT Bombay) To jump the queue or wait my turn? January 14, 2014 12 / 24

A Theorem For 0 ≤ t ≤ 1 and a ≤ v 1 ≤ b define  0 for v > v 1 ,   p E ( v ) = t for v = v 1 ,  1 for v < v 1 .  � v 2 λ 2 W 0 y X E ( v ) = (1 − ρ 2 − ρ 2 F 1 ( y )) 3 dF 1 ( y ) 0 ( p E ( v ) , X E ( v )) is a stable policy with v 1 as above. D. Manjunath (IIT Bombay) To jump the queue or wait my turn? January 14, 2014 12 / 24

A Theorem For 0 ≤ t ≤ 1 and a ≤ v 1 ≤ b define  0  v ≤ v 1 0 for v > v 1 ,    F ( v )   p E ( v ) = v 1 ≤ v ≤ b , F 1 ( v ) := t for v = v 1 , � b v 1 dF ( v )   1 for v < v 1 .   1 v > b .  � v 2 λ 2 W 0 y X E ( v ) = (1 − ρ 2 − ρ 2 F 1 ( y )) 3 dF 1 ( y ) 0 ( p E ( v ) , X E ( v )) is a stable policy with v 1 as above. D. Manjunath (IIT Bombay) To jump the queue or wait my turn? January 14, 2014 12 / 24

A Theorem For 0 ≤ t ≤ 1 and a ≤ v 1 ≤ b define  0  v ≤ v 1 0 for v > v 1 ,    F ( v )   p E ( v ) = v 1 ≤ v ≤ b , F 1 ( v ) := t for v = v 1 , � b v 1 dF ( v )   1 for v < v 1 .   1 v > b .  � v 2 λ 2 W 0 y X E ( v ) = (1 − ρ 2 − ρ 2 F 1 ( y )) 3 dF 1 ( y ) 0 Let W 1 be expected delay in FIFO queue with arrival rate λ 1 and W 2 ( v ) expected delay in bidding queue with bidding function X E ( v ) . ( p E ( v ) , X E ( v )) is a stable policy with v 1 as above. D. Manjunath (IIT Bombay) To jump the queue or wait my turn? January 14, 2014 12 / 24

A Theorem For 0 ≤ t ≤ 1 and a ≤ v 1 ≤ b define  0  v ≤ v 1 0 for v > v 1 ,    F ( v )   p E ( v ) = v 1 ≤ v ≤ b , F 1 ( v ) := t for v = v 1 , � b v 1 dF ( v )   1 for v < v 1 .   1 v > b .  � v 2 λ 2 W 0 y X E ( v ) = (1 − ρ 2 − ρ 2 F 1 ( y )) 3 dF 1 ( y ) 0 Let W 1 be expected delay in FIFO queue with arrival rate λ 1 and W 2 ( v ) expected delay in bidding queue with bidding function X E ( v ) . Let v 1 satisfy v 1 W 2 ( v 1 ) = v 1 W 1 − M . ( p E ( v ) , X E ( v )) is a stable policy with v 1 as above. D. Manjunath (IIT Bombay) To jump the queue or wait my turn? January 14, 2014 12 / 24

Proof Sketch For v < v 1 , we have M = v 1 ( W 1 − W 2 ( v 1 )) > v ( W 1 − W 2 ( v 1 )) This implies vW 1 − M < vW 2 ( v 1 )) and hence for v < v 1 , we have p E ( v ) = 1; D. Manjunath (IIT Bombay) To jump the queue or wait my turn? January 14, 2014 13 / 24

Proof Sketch For v < v 1 , we have M = v 1 ( W 1 − W 2 ( v 1 )) > v ( W 1 − W 2 ( v 1 )) This implies vW 1 − M < vW 2 ( v 1 )) and hence for v < v 1 , we have p E ( v ) = 1; For v > v 1 , using concavity of C ( v ) and the property that dC ( v ) = W ( x ) it can be shown that for ǫ > 0 , dv C ( v 1 + ǫ ) < ( v 1 + ǫ ) W 1 − M implying that p E ( v ) = 0 . D. Manjunath (IIT Bombay) To jump the queue or wait my turn? January 14, 2014 13 / 24

Discussion v 1 satisfying v 1 W 2 ( v 1 ) = v 1 W 1 − M is unique. D. Manjunath (IIT Bombay) To jump the queue or wait my turn? January 14, 2014 14 / 24

Discussion v 1 satisfying v 1 W 2 ( v 1 ) = v 1 W 1 − 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 D. Manjunath (IIT Bombay) To jump the queue or wait my turn? January 14, 2014 14 / 24

Discussion v 1 satisfying v 1 W 2 ( v 1 ) = v 1 W 1 − 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 . v 1 W 2 ( v 1 ) = v 1 W 1 − M v 1 W 2 ( v 1 ) + M = v 1 W 1 D. Manjunath (IIT Bombay) To jump the queue or wait my turn? January 14, 2014 14 / 24

Discussion (contd.) For a fixed λ, F ( v ) and µ 2 , the revenue rate at equilibrium decreases with increasing µ 1 . D. Manjunath (IIT Bombay) To jump the queue or wait my turn? January 14, 2014 15 / 24

Discussion (contd.) For a fixed λ, F ( v ) and µ 2 , the revenue rate at equilibrium decreases with increasing µ 1 . Decreasing λ decreases revenue rate. D. Manjunath (IIT Bombay) To jump the queue or wait my turn? January 14, 2014 15 / 24

Discussion (contd.) For a fixed λ, F ( v ) and µ 2 , the revenue rate at equilibrium decreases with increasing µ 1 . Decreasing λ decreases revenue rate. Recall that the left truncation of the sensitivity function increases the revenue rate in a single server queue. D. Manjunath (IIT Bombay) To jump the queue or wait my turn? January 14, 2014 15 / 24

Discussion (contd.) For a fixed λ, F ( v ) and µ 2 , the revenue rate at equilibrium decreases with increasing µ 1 . Decreasing λ decreases revenue rate. Recall that the left truncation of the sensitivity function increases the revenue rate in a single server queue. This leads us to ask the following question. D. Manjunath (IIT Bombay) To jump the queue or wait my turn? January 14, 2014 15 / 24

Discussion (contd.) For a fixed λ, F ( v ) and µ 2 , the revenue rate at equilibrium decreases with increasing µ 1 . Decreasing λ decreases revenue rate. Recall that the left truncation of the sensitivity function increases the revenue rate in a single server queue. This leads us to ask the following question. Assume that a total capacity of µ is available D. Manjunath (IIT Bombay) To jump the queue or wait my turn? January 14, 2014 15 / 24

Discussion (contd.) For a fixed λ, F ( v ) and µ 2 , the revenue rate at equilibrium decreases with increasing µ 1 . Decreasing λ decreases revenue rate. Recall that the left truncation of the sensitivity function increases the revenue 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, µ 1 is allocated to the FIFO queue and µ 2 = µ − µ 1 is allocated to the bidding queue. D. Manjunath (IIT Bombay) To jump the queue or wait my turn? January 14, 2014 15 / 24

Discussion (contd.) For a fixed λ, F ( v ) and µ 2 , the revenue rate at equilibrium decreases with increasing µ 1 . Decreasing λ decreases revenue rate. Recall that the left truncation of the sensitivity function increases the revenue 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, µ 1 is allocated to the FIFO queue and µ 2 = µ − µ 1 is allocated to the bidding queue. How does revenue vary as a function of µ 1 ? D. Manjunath (IIT Bombay) To jump the queue or wait my turn? January 14, 2014 15 / 24

Numerical Results v Example 1: λ = 0 . 5 , µ = 1 F ( v ) = 100 for 0 ≤ v ≤ 100 . Example 2: λ = 0 . 9 , µ = 1 F ( v ) = v 0 . 5 10 for 0 ≤ v ≤ 100 . Revenue as a function of µ 2 with M = 0 D. Manjunath (IIT Bombay) To jump the queue or wait my turn? January 14, 2014 16 / 24

Numerical Results v Example 1: λ = 0 . 5 , µ = 1 F ( v ) = 100 for 0 ≤ v ≤ 100 . Example 2: λ = 0 . 9 , µ = 1 F ( v ) = v 0 . 5 10 for 0 ≤ v ≤ 100 . Revenue as a function of µ 2 with M = 0 160 140 120 100 Revenue 80 60 40 20 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 µ 2 Example 1 D. Manjunath (IIT Bombay) To jump the queue or wait my turn? January 14, 2014 16 / 24

Numerical Results v Example 1: λ = 0 . 5 , µ = 1 F ( v ) = 100 for 0 ≤ v ≤ 100 . Example 2: λ = 0 . 9 , µ = 1 F ( v ) = v 0 . 5 10 for 0 ≤ v ≤ 100 . Revenue as a function of µ 2 with M = 0 2500 160 140 2000 120 100 1500 Revenue Revenue 80 60 1000 40 500 20 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 µ 2 0 0.2 0.4 µ 2 0.6 0.8 1 Example 1 Example 2 D. Manjunath (IIT Bombay) To jump the queue or wait my turn? January 14, 2014 16 / 24

Numerical Results (contd.) v Example 1: λ = 0 . 5 , µ = 1 F ( v ) = 100 for 0 ≤ v ≤ 100 . Example 2: λ = 0 . 9 , µ = 1 F ( v ) = v 0 . 5 10 for 0 ≤ v ≤ 100 . Threshold as a function of M for µ 1 = 0 . 5 . D. Manjunath (IIT Bombay) To jump the queue or wait my turn? January 14, 2014 17 / 24

Numerical Results (contd.) v Example 1: λ = 0 . 5 , µ = 1 F ( v ) = 100 for 0 ≤ v ≤ 100 . Example 2: λ = 0 . 9 , µ = 1 F ( v ) = v 0 . 5 10 for 0 ≤ v ≤ 100 . Threshold as a function of M for µ 1 = 0 . 5 . 100 95 90 85 v 1 80 75 70 65 60 0 1000 2000 M 3000 4000 5000 Example 1 D. Manjunath (IIT Bombay) To jump the queue or wait my turn? January 14, 2014 17 / 24

Numerical Results (contd.) v Example 1: λ = 0 . 5 , µ = 1 F ( v ) = 100 for 0 ≤ v ≤ 100 . Example 2: λ = 0 . 9 , µ = 1 F ( v ) = v 0 . 5 10 for 0 ≤ v ≤ 100 . Threshold as a function of M for µ 1 = 0 . 5 . 100 31 95 30.5 90 85 v 1 v 1 30 80 75 29.5 70 65 29 60 0 2 4 6 8 10 M 0 1000 2000 M 3000 4000 5000 4 x 10 Example 1 Example 2 D. Manjunath (IIT Bombay) To jump the queue or wait my turn? January 14, 2014 17 / 24

Numerical Results (contd.) v Example 1: λ = 0 . 5 , µ = 1 F ( v ) = 100 for 0 ≤ v ≤ 100 . Example 2: λ = 0 . 9 , µ = 1 F ( v ) = v 0 . 5 10 for 0 ≤ v ≤ 100 . Let µ 2 , min be the minimum service rate required to earn positive revenue. µ 2 , min as a function of M . D. Manjunath (IIT Bombay) To jump the queue or wait my turn? January 14, 2014 18 / 24

Numerical Results (contd.) v Example 1: λ = 0 . 5 , µ = 1 F ( v ) = 100 for 0 ≤ v ≤ 100 . Example 2: λ = 0 . 9 , µ = 1 F ( v ) = v 0 . 5 10 for 0 ≤ v ≤ 100 . Let µ 2 , min be the minimum service rate required to earn positive revenue. µ 2 , min as a function of M . 0.5 0.45 µ 2 Minimum 0.4 0.35 0.3 0.25 0 500 1000 1500 2000 M Example 1 D. Manjunath (IIT Bombay) To jump the queue or wait my turn? January 14, 2014 18 / 24

Numerical Results (contd.) v Example 1: λ = 0 . 5 , µ = 1 F ( v ) = 100 for 0 ≤ v ≤ 100 . Example 2: λ = 0 . 9 , µ = 1 F ( v ) = v 0 . 5 10 for 0 ≤ v ≤ 100 . Let µ 2 , min be the minimum service rate required to earn positive revenue. µ 2 , min as a function of M . 0.1 0.5 0.095 0.45 0.09 0.085 µ 2 minimum µ 2 Minimum 0.4 0.08 0.075 0.35 0.07 0.065 0.3 0.06 0.25 0 2000 4000 6000 8000 10000 0 500 1000 1500 2000 M M Example 1 Example 2 D. Manjunath (IIT Bombay) To jump the queue or wait my turn? January 14, 2014 18 / 24

Lessons For a fixed µ, the revenue can actually increase if some capacity is allocated to a FIFO queue. D. Manjunath (IIT Bombay) To jump the queue or wait my turn? January 14, 2014 19 / 24

Lessons For a fixed µ, the revenue can actually increase if some capacity is allocated to a FIFO queue. If µ 1 < λ then some arrivals will choose the bidding queue and there will be non zero revenue. D. Manjunath (IIT Bombay) To jump the queue or wait my turn? January 14, 2014 19 / 24

Lessons For a fixed µ, the revenue can actually increase if some capacity is allocated to a FIFO queue. If µ 1 < λ then some arrivals will choose the bidding queue and there will be non zero revenue. However, if µ 1 > λ, then for sufficiently large M , there can be no revenue. D. Manjunath (IIT Bombay) To jump the queue or wait my turn? January 14, 2014 19 / 24

Lessons For a fixed µ, the revenue can actually increase if some capacity is allocated to a FIFO queue. If µ 1 < λ then some arrivals will choose the bidding queue and there will be non zero revenue. However, if µ 1 > λ, then for sufficiently large M , there can be no revenue. Now interpret the server as a babu (public service official(?)), the bids as bribes, M as society’s reward to the customers for choosing the righteous way. D. Manjunath (IIT Bombay) To jump the queue or wait my turn? January 14, 2014 19 / 24

Lessons For a fixed µ, the revenue can actually increase if some capacity is allocated to a FIFO queue. If µ 1 < λ then some arrivals will choose the bidding queue and there will be non zero revenue. However, if µ 1 > λ, then for sufficiently large M , there can be no revenue. Now interpret the server as a babu (public service official(?)), the bids as bribes, M as society’s reward to the customers for choosing the righteous way. There is more money to be made by being partially honest than being fully dishonest. D. Manjunath (IIT Bombay) To jump the queue or wait my turn? January 14, 2014 19 / 24

Lessons For a fixed µ, the revenue can actually increase if some capacity is allocated to a FIFO queue. If µ 1 < λ then some arrivals will choose the bidding queue and there will be non zero revenue. However, if µ 1 > λ, then for sufficiently large M , there can be no revenue. Now interpret the server as a babu (public service official(?)), the bids as bribes, M as society’s reward to the customers for choosing the righteous way. There is more money to be made by being partially honest than being fully dishonest. Of course, the latter conclusions are in jest and not meant to be used in a sociological study. D. Manjunath (IIT Bombay) To jump the queue or wait my turn? January 14, 2014 19 / 24

Overtime you say? Now consider what happens if we add capacity to the FIFO queue without affecting the bidding queue. D. Manjunath (IIT Bombay) To jump the queue or wait my turn? January 14, 2014 20 / 24

Overtime you say? Now consider what happens if we add capacity to the FIFO queue without affecting the bidding queue. Specifically, we study the revenue at equilibrium as a function of µ 1 with µ 2 and all other parameters ( M , F ( v ) , and λ ) being fixed. D. Manjunath (IIT Bombay) To jump the queue or wait my turn? January 14, 2014 20 / 24

Overtime you say? Now consider what happens if we add capacity to the FIFO queue without affecting the bidding queue. Specifically, we study the revenue at equilibrium as a function of µ 1 with µ 2 and all other parameters ( M , F ( v ) , and λ ) being fixed. Notation is simplified to R ( µ 1 ) for the revenue rate as a function of µ 1 . D. Manjunath (IIT Bombay) To jump the queue or wait my turn? January 14, 2014 20 / 24

Overtime you say? Now consider what happens if we add capacity to the FIFO queue without affecting the bidding queue. Specifically, we study the revenue at equilibrium as a function of µ 1 with µ 2 and all other parameters ( M , F ( v ) , and λ ) being fixed. Notation is simplified to R ( µ 1 ) for the revenue rate as a function of µ 1 . Lemma R (0) ≥ R ( µ 1 ) i.e., adding capacity in the form of FIFO queue does not increase the revenue. D. Manjunath (IIT Bombay) To jump the queue or wait my turn? January 14, 2014 20 / 24

Giving Up: The Balking Case Balking : If the service provided has a value, say P , then the customers whose total cost C ( v ) exceeds P prefer not to take the service. D. Manjunath (IIT Bombay) To jump the queue or wait my turn? January 14, 2014 21 / 24

Giving Up: The Balking Case Balking : If the service provided has a value, say P , then the customers whose total cost C ( v ) exceeds P prefer not to take the service. In such a balking model P = X ( v ∗ ) + v ∗ W ( v ∗ ) where v ∗ is the highest sensitivity of a customer joining the queue and customers with v > v ∗ balk. D. Manjunath (IIT Bombay) To jump the queue or wait my turn? January 14, 2014 21 / 24

Giving Up: The Balking Case Balking : If the service provided has a value, say P , then the customers whose total cost C ( v ) exceeds P prefer not to take the service. In such a balking model P = X ( v ∗ ) + v ∗ W ( v ∗ ) where v ∗ is the highest sensitivity of a customer joining the queue and customers with v > v ∗ balk. Let v ∗ ( µ 1 ) be the highest sensitivity of customers joining the bidding queue when the service rate of the FIFO queue is µ 1 . D. Manjunath (IIT Bombay) To jump the queue or wait my turn? January 14, 2014 21 / 24

Results Theorem For uniform sensitivity profile F ( v ) = Av , and fixed µ 2 , D. Manjunath (IIT Bombay) To jump the queue or wait my turn? January 14, 2014 22 / 24

Results Theorem For uniform sensitivity profile F ( v ) = Av , and fixed µ 2 , v ∗ ( µ 1 ) increases with µ 1 . D. Manjunath (IIT Bombay) To jump the queue or wait my turn? January 14, 2014 22 / 24

Results Theorem For uniform sensitivity profile F ( v ) = Av , and fixed µ 2 , v ∗ ( µ 1 ) increases with µ 1 . R ( µ 1 ) decreases with µ 1 , . i.e., adding a FIFO queue decreases revenue. D. Manjunath (IIT Bombay) To jump the queue or wait my turn? January 14, 2014 22 / 24

Concluding Remarks A motivation Primary motivation came from developing bidding models for service systems with queues and arrivals. D. Manjunath (IIT Bombay) To jump the queue or wait my turn? January 14, 2014 23 / 24

Concluding Remarks A motivation Primary motivation came from developing bidding models for service systems with queues and arrivals. For such a scenario, an obvious variation is to have processor sharing and allocate the weights in proportion to the bids. D. Manjunath (IIT Bombay) To jump the queue or wait my turn? January 14, 2014 23 / 24

Concluding Remarks A motivation Primary motivation came from developing bidding models for service systems with queues and arrivals. For such a scenario, an obvious variation is to have processor sharing and allocate the weights in proportion to the bids. A secondary motivation is to explore new models to test Myrdal’s claim that corruption in public service decreases efficiency. D. Manjunath (IIT Bombay) To jump the queue or wait my turn? January 14, 2014 23 / 24

Concluding Remarks A motivation Primary motivation came from developing bidding models for service systems with queues and arrivals. For such a scenario, an obvious variation is to have processor sharing and allocate the weights in proportion to the bids. A secondary motivation is to explore new models to test Myrdal’s claim that corruption in public service decreases efficiency. In this context, M was a simplistic model for reward. Other reward structures are being explored. D. Manjunath (IIT Bombay) To jump the queue or wait my turn? January 14, 2014 23 / 24

Acknowledgements The first part is a summary from D. Manjunath (IIT Bombay) To jump the queue or wait my turn? January 14, 2014 24 / 24

Acknowledgements The first part is a summary from L. Kleinrock, “Optimal bribing for queue position,” Operations Research, vol 15, pp. 304–318, 1967. D. Manjunath (IIT Bombay) To jump the queue or wait my turn? January 14, 2014 24 / 24

Acknowledgements The first part is a summary from L. Kleinrock, “Optimal bribing for queue position,” Operations Research, vol 15, pp. 304–318, 1967. F. Lui, “An equilibrium model of bribery,” Political Economy, vol. 93, pp. 760–781, 1985. D. Manjunath (IIT Bombay) To jump the queue or wait my turn? January 14, 2014 24 / 24