Regulating Arrivals to a Queue When Customers Know their Demand - PowerPoint PPT Presentation

Regulating Arrivals to a Queue When Customers Know their Demand Moshe Haviv Department of Statistics and Center for the Study of Rationality The Hebrew University of Jerusalem Brisbane, July 2013 1 / 57

The basic queueing model (M/M/1) single server first come first served (FCFS) Poisson arrival rate λ exponential service rate µ > λ (mean of 1 µ ) value of service R cost per unit of wait C 2 / 57

Some facts mean service time 1 /µ utilization level ρ = λ/µ < 1 mean time in the system 1 W = µ (1 − ρ ) 3 / 57

Some facts mean service time 1 /µ utilization level ρ = λ/µ < 1 mean time in the system 1 W = µ (1 − ρ ) mean time in the system for a stand-by customer 1 µ (1 − ρ ) 2 equals the total added time to the society due to the marginal arrival 4 / 57

Some facts mean service time 1 /µ utilization level ρ = λ/µ < 1 mean time in the system 1 W = µ (1 − ρ ) mean time in the system for a stand-by customer 1 µ (1 − ρ ) 2 equals the total added time to the society due to the marginal arrival Example: assume λ = 0 . 9 and 1 /µ = 1 ⇒ ρ = 0 . 9 ⇒ mean time in the system 10 ⇒ mean socially added time 100 (for 1 unit of service!) 5 / 57

Some facts mean service time 1 /µ utilization level ρ = λ/µ < 1 mean time in the system 1 W = µ (1 − ρ ) mean time in the system for a stand-by customer 1 µ (1 − ρ ) 2 equals the total added time to the society due to the marginal arrival Example: assume λ = 0 . 9 and 1 /µ = 1 ⇒ ρ = 0 . 9 ⇒ mean time in the system 10 ⇒ mean socially added time 100 (for 1 unit of service!) You care for the 10, not for the 100. This is why queues are too long. 6 / 57

To queue or not to queue Edleson and Hildebrand, ‘75 assume R − C C µ > 0 and R − µ (1 − ρ ) < 0 if nobody joins, one better joins. If all join, one better do not join. 7 / 57

To queue or not to queue Edleson and Hildebrand, ‘75 assume R − C C µ > 0 and R − µ (1 − ρ ) < 0 if nobody joins, one better joins. If all join, one better do not join. (Nash) equilibrium: join with probability p e where C R − µ (1 − p e ρ ) = 0 In equilibrium, all are indifferent between joining or not. 8 / 57

To queue or not to queue Edleson and Hildebrand, ‘75 assume R − C C µ > 0 and R − µ (1 − ρ ) < 0 if nobody joins, one better joins. If all join, one better do not join. (Nash) equilibrium: join with probability p e where C R − µ (1 − p e ρ ) = 0 In equilibrium, all are indifferent between joining or not. social optimization: join with probability p s where C p s = arg max 0 < p < p e p λ ( R − µ (1 − p ρ )) 9 / 57

To queue or not to queue Edleson and Hildebrand, ‘75 assume R − C C µ > 0 and R − µ (1 − ρ ) < 0 if nobody joins, one better joins. If all join, one better do not join. (Nash) equilibrium: join with probability p e where C R − µ (1 − p e ρ ) = 0 In equilibrium, all are indifferent between joining or not. social optimization: join with probability p s where C p s = arg max 0 < p < p e p λ ( R − µ (1 − p ρ )) C R − µ (1 − p s ρ ) 2 = 0 In social optimization, the society is indifferent whether the marginal customer joins or not. 10 / 57

Some facts The equilibrium arrival rate: λ e = µ − C R . � C µ The socially optimal arrival rate: λ s = µ − R . Either rate is not a function of the potential rate. λ s < λ e ⇒ long queues The consumer surplus is zero in equilibrium. √ It is ( √ R µ − C ) 2 in social optimization. 11 / 57

Regulating by an entry fee (Pigouvian tax) socially optimal entry fee T : C R − T − µ (1 − p s ρ ) = 0 ⇓ � CR T = R − CW = R − µ 12 / 57

Regulating by an entry fee (Pigouvian tax) socially optimal entry fee T : C R − T − µ (1 − p s ρ ) = 0 ⇓ � CR T = R − CW = R − µ C C T = µ (1 − p s ρ ) 2 − µ (1 − p s ρ ) T = externalities the marginal joiner inflicts under the socially optimal scenario 13 / 57

Waiting cost marginal social cost individual cost R R − T p s p e p 14 / 57

Regulating by increasing waiting costs the same effect is achieved with an added holding fee h : C + h R − µ (1 − p s ρ ) = 0 ⇓ � h = RC µ − C 15 / 57

Regulating contracts A contract: if you join, pay f ( X ) for some unknown random variable X. If E( f ( X )) coincides with the externalities under social optimal joining rate, this scheme leads to regulation. f ( X ) = the expected externalities given X . 16 / 57

Regulating contracts A contract: if you join, pay f ( X ) for some unknown random variable X. If E( f ( X )) coincides with the externalities under social optimal joining rate, this scheme leads to regulation. f ( X ) = the expected externalities given X . Possible random variables: time in the system queue length upon arrival queue length upon departure service time 17 / 57

Expected Externalities W = time in the system (service inclusive) �� � λ s W R µ C µ (1 − p s ρ ) = C C − 1 W 18 / 57

Expected Externalities W = time in the system (service inclusive) �� � λ s W R µ C µ (1 − p s ρ ) = C C − 1 W L a = number in the system upon arrival (inclusive) �� � µ (1 − p s ρ ) − C L a L a C µ − 1 R µ = C C L a µ 19 / 57

Expected Externalities W = time in the system (service inclusive) �� � λ s W R µ C µ (1 − p s ρ ) = C C − 1 W L a = number in the system upon arrival (inclusive) �� � µ (1 − p s ρ ) − C L a L a C µ − 1 R µ = C C L a µ L d = number in the system upon departure (exclusive) � C CR µ (1 − p s ρ ) L d = µ L d 20 / 57

Expected Externalities W = time in the system (service inclusive) �� � λ s W R µ C µ (1 − p s ρ ) = C C − 1 W L a = number in the system upon arrival (inclusive) �� � µ (1 − p s ρ ) − C L a L a C µ − 1 R µ = C C L a µ L d = number in the system upon departure (exclusive) � C CR µ (1 − p s ρ ) L d = µ L d S = service time ( p s ρ ) 2 λ s 2(1 − p s ρ ) S 2 + C C (1 − p s ρ ) 2 S 21 / 57

Quadratic fees Kelly, ’91 W = waiting time Charge aW 2 + bW . Any a , b with a E( W 2 ) + b E( W ) = T will do For example, a = C µ/ 2 and b = − 1 22 / 57

Quadratic fees Kelly, ’91 W = waiting time Charge aW 2 + bW . Any a , b with a E( W 2 ) + b E( W ) = T will do For example, a = C µ/ 2 and b = − 1 These a and b are free of R ! This is the unique function f ( W ) with E( f ( W )) = T which is free of R A similar scheme with L a 23 / 57

Some facts customers internalize the externalities they inflict on others 24 / 57

Some facts customers internalize the externalities they inflict on others all the consumer surplus goes to the central planner √ � C ) 2 ( R µ − 25 / 57

Some facts customers internalize the externalities they inflict on others all the consumer surplus goes to the central planner √ � C ) 2 ( R µ − customers are ending up with nothing as they possess no private information 26 / 57

C + h C C Waiting cost µ (1 − p ρ )2 µ (1 − p ρ ) µ (1 − p ρ (2) (3) (1) (1) individual cost (2) marginal social cost (3) holding cost R R − T p s p e 27 / 57

Regulating by pessimism p e equilibrium joining probability p s socially optimal joining probability 28 / 57

Regulating by pessimism p e equilibrium joining probability p s socially optimal joining probability 1 1 Interestingly, µ (1 − p e ρ ) = µ (1 − p s ρ ) 2 and hence, C R − µ (1 − p s ρ ) 2 = 0 Under a socially optimal joining probability, a stand-by customer is indifferent between joining or not. So is the society: He inflicts no externalities. But society does not mind order of service 29 / 57

Regulating by pessimism p e equilibrium joining probability p s socially optimal joining probability 1 1 Interestingly, µ (1 − p e ρ ) = µ (1 − p s ρ ) 2 and hence, C R − µ (1 − p s ρ ) 2 = 0 Under a socially optimal joining probability, a stand-by customer is indifferent between joining or not. So is the society: He inflicts no externalities. But society does not mind order of service If all think they are stand-by customers, then p s is an equilibrium. Problem: contradicts standard assumptions in games and economics: all being last cannot be common knowledge.... 30 / 57

When customers know their demand M/G/1, g ( x ) density of service time customers know their demand and decide whether to join or not 31 / 57

When customers know their demand M/G/1, g ( x ) density of service time customers know their demand and decide whether to join or not W x ( y )= mean time for a y job, when x is the threshold L x = mean number in the system assumption: some threshold strategy is a best response 32 / 57

When customers know their demand M/G/1, g ( x ) density of service time customers know their demand and decide whether to join or not W x ( y )= mean time for a y job, when x is the threshold L x = mean number in the system assumption: some threshold strategy is a best response equilibrium threshold: R − CW x e ( x e ) = 0 x e is a best response against x e . 33 / 57

socially optimal threshold: x s = arg max x { λ G ( x ) R − CL x } 34 / 57