Rest in Lounge or Wait in Queue
Sandeep Juneja, TIFR
Visiting Professor CAFRAL, Central Bank India
.
initial parts are joint work with
Nahum Shimkin (Technion, Israel), Rahul Jain (USC),
.
latter with D. Manjunath (IITB)
.
Workshop on Congestion Games, NUS
December 16, 2015
Workshop on Congestion Gam
Sandeep Juneja, TIFR Visiting Professor
Strategic Arrivals
CAFRAL,
to Queues Central Bank India . initial parts are /joint
43 w
Queue timing games considered
Workshop on Congestion Gam
Sandeep Juneja, TIFR Visiting Professor
Strategic Arrivals
CAFRAL,
to Queues Central Bank India . initial parts are /joint
43 w
Torestinloungeorwaitinqueue
Concertqueue,queuesizenotknown
Workshop on Congestion Gam
Sandeep Juneja, TIFR Visiting Professor
Strategic Arrivals
CAFRAL,
to Queues Central Bank India . initial parts are /joint
43 w
Torestinloungeorwaitinqueue
Concertqueue,queuesizeknown
Workshop on Congestion Gam
Sandeep Juneja, TIFR Visiting Professor
Strategic Arrivals
CAFRAL,
to Queues Central Bank India . initial parts are /joint
43 w
Torestinloungeorwaitinqueue
M/M/1queue,queuesizeknown
Workshop on Congestion Gam
Sandeep Juneja, TIFR Visiting Professor
Strategic Arrivals
CAFRAL,
to Queues Central Bank India . initial parts are /joint
43 w
Related Literature: Strategic Timing Decisions
Substantial transportation literature motivated by Vickrey’s
seminal work on the morning commute problem:
Drivers going to work via single bottleneck queue. They have a
preferred time to reach office. Too early or too late not good.
Vehicles modelled as fluid particles
Focus primarily on coming up with toll regimes to manage congestion.
Earlier queueing literature focusses primarily on admissions
control, routing, reneging, pricing etc. Largely summarized in
monograph ‘To Queue or not to Queue’ by Hassin and Haviv
(2003).
Workshop on Congestion Gam
Sandeep Juneja, TIFR Visiting Professor
Strategic Arrivals
CAFRAL,
to Queues Central Bank India . initial parts are /joint
43 w
Related Literature: Strategic Timing Decisions
Substantial transportation literature motivated by Vickrey’s
seminal work on the morning commute problem:
Drivers going to work via single bottleneck queue. They have a
preferred time to reach office. Too early or too late not good.
Vehicles modelled as fluid particles
Focus primarily on coming up with toll regimes to manage congestion.
Earlier queueing literature focusses primarily on admissions
control, routing, reneging, pricing etc. Largely summarized in
monograph ‘To Queue or not to Queue’ by Hassin and Haviv
(2003).
Workshop on Congestion Gam
Sandeep Juneja, TIFR Visiting Professor
Strategic Arrivals
CAFRAL,
to Queues Central Bank India . initial parts are /joint
43 w
Related Literature: Strategic Timing Decisions
Substantial transportation literature motivated by Vickrey’s
seminal work on the morning commute problem:
Drivers going to work via single bottleneck queue. They have a
preferred time to reach office. Too early or too late not good.
Vehicles modelled as fluid particles
Focus primarily on coming up with toll regimes to manage congestion.
Earlier queueing literature focusses primarily on admissions
control, routing, reneging, pricing etc. Largely summarized in
monograph ‘To Queue or not to Queue’ by Hassin and Haviv
(2003).
Workshop on Congestion Gam
Sandeep Juneja, TIFR Visiting Professor
Strategic Arrivals
CAFRAL,
to Queues Central Bank India . initial parts are /joint
43 w
Related Literature: Strategic Timing Decisions
Substantial transportation literature motivated by Vickrey’s
seminal work on the morning commute problem:
Drivers going to work via single bottleneck queue. They have a
preferred time to reach office. Too early or too late not good.
Vehicles modelled as fluid particles
Focus primarily on coming up with toll regimes to manage congestion.
Earlier queueing literature focusses primarily on admissions
control, routing, reneging, pricing etc. Largely summarized in
monograph ‘To Queue or not to Queue’ by Hassin and Haviv
(2003).
Workshop on Congestion Gam
Sandeep Juneja, TIFR Visiting Professor
Strategic Arrivals
CAFRAL,
to Queues Central Bank India . initial parts are /joint
43 w
Related Literature: Strategic Timing Decisions
Substantial transportation literature motivated by Vickrey’s
seminal work on the morning commute problem:
Drivers going to work via single bottleneck queue. They have a
preferred time to reach office. Too early or too late not good.
Vehicles modelled as fluid particles
Focus primarily on coming up with toll regimes to manage congestion.
Earlier queueing literature focusses primarily on admissions
control, routing, reneging, pricing etc. Largely summarized in
monograph ‘To Queue or not to Queue’ by Hassin and Haviv
(2003).
Workshop on Congestion Gam
Sandeep Juneja, TIFR Visiting Professor
Strategic Arrivals
CAFRAL,
to Queues Central Bank India . initial parts are /joint
43 w
Concert or Cafeteria Queueing Problem
When to arrive at a musical concert? Cafeteria for lunch?
Passport Office? Board an airplane? Movie theater? doctor’s
clinic?
Too early may mean large queues and long wait.
Too late means that one gets poorer service, gets served late
In this talk,
1
2
3
we review the literature in fluid framework when the total population is
fixed
we also discuss the finite population case,
as well as the dynamic setting where people in ‘lounge’ can observe
queue lengths and decide when to join.
Workshop on Congestion Gam
Sandeep Juneja, TIFR Visiting Professor
Strategic Arrivals
CAFRAL,
to Queues Central Bank India . initial parts are /joint
43 w
Concert or Cafeteria Queueing Problem
When to arrive at a musical concert? Cafeteria for lunch?
Passport Office? Board an airplane? Movie theater? doctor’s
clinic?
Too early may mean large queues and long wait.
Too late means that one gets poorer service, gets served late
In this talk,
1
2
3
we review the literature in fluid framework when the total population is
fixed
we also discuss the finite population case,
as well as the dynamic setting where people in ‘lounge’ can observe
queue lengths and decide when to join.
Workshop on Congestion Gam
Sandeep Juneja, TIFR Visiting Professor
Strategic Arrivals
CAFRAL,
to Queues Central Bank India . initial parts are /joint
43 w
Concert or Cafeteria Queueing Problem
When to arrive at a musical concert? Cafeteria for lunch?
Passport Office? Board an airplane? Movie theater? doctor’s
clinic?
Too early may mean large queues and long wait.
Too late means that one gets poorer service, gets served late
In this talk,
1
2
3
we review the literature in fluid framework when the total population is
fixed
we also discuss the finite population case,
as well as the dynamic setting where people in ‘lounge’ can observe
queue lengths and decide when to join.
Workshop on Congestion Gam
Sandeep Juneja, TIFR Visiting Professor
Strategic Arrivals
CAFRAL,
to Queues Central Bank India . initial parts are /joint
43 w
Concert or Cafeteria Queueing Problem
When to arrive at a musical concert? Cafeteria for lunch?
Passport Office? Board an airplane? Movie theater? doctor’s
clinic?
Too early may mean large queues and long wait.
Too late means that one gets poorer service, gets served late
In this talk,
1
2
3
we review the literature in fluid framework when the total population is
fixed
we also discuss the finite population case,
as well as the dynamic setting where people in ‘lounge’ can observe
queue lengths and decide when to join.
Workshop on Congestion Gam
Sandeep Juneja, TIFR Visiting Professor
Strategic Arrivals
CAFRAL,
to Queues Central Bank India . initial parts are /joint
43 w
Concert or Cafeteria Queueing Problem
When to arrive at a musical concert? Cafeteria for lunch?
Passport Office? Board an airplane? Movie theater? doctor’s
clinic?
Too early may mean large queues and long wait.
Too late means that one gets poorer service, gets served late
In this talk,
1
2
3
we review the literature in fluid framework when the total population is
fixed
we also discuss the finite population case,
as well as the dynamic setting where people in ‘lounge’ can observe
queue lengths and decide when to join.
Workshop on Congestion Gam
Sandeep Juneja, TIFR Visiting Professor
Strategic Arrivals
CAFRAL,
to Queues Central Bank India . initial parts are /joint
43 w
Concert or Cafeteria Queueing Problem
When to arrive at a musical concert? Cafeteria for lunch?
Passport Office? Board an airplane? Movie theater? doctor’s
clinic?
Too early may mean large queues and long wait.
Too late means that one gets poorer service, gets served late
In this talk,
1
2
3
we review the literature in fluid framework when the total population is
fixed
we also discuss the finite population case,
as well as the dynamic setting where people in ‘lounge’ can observe
queue lengths and decide when to join.
Workshop on Congestion Gam
Sandeep Juneja, TIFR Visiting Professor
Strategic Arrivals
CAFRAL,
to Queues Central Bank India . initial parts are /joint
43 w
Torestinloungeorwaitinqueue
Concertqueue,queuesizenotknown
Workshop on Congestion Gam
Sandeep Juneja, TIFR Visiting Professor
Strategic Arrivals
CAFRAL,
to Queues Central Bank India . initial parts are /joint
43 w
Fluid model framework
Fluid model
Fluid model where arrivals are points in a continuum [0, 1], server
serves at a fixed rate µ.
Server becomes active at time zero
Customers can come and queue up before or after time zero
Customer cost is additive and linear in waiting time and service
completion time (equivalently, in time spent in lounge)
Workshop on Congestion Gam
Sandeep Juneja, TIFR Visiting Professor
Strategic Arrivals
CAFRAL,
to Queues Central Bank India . initial parts are /joint
43 w
Fluid model framework
Fluid model
Fluid model where arrivals are points in a continuum [0, 1], server
serves at a fixed rate µ.
Server becomes active at time zero
Customers can come and queue up before or after time zero
Customer cost is additive and linear in waiting time and service
completion time (equivalently, in time spent in lounge)
Workshop on Congestion Gam
Sandeep Juneja, TIFR Visiting Professor
Strategic Arrivals
CAFRAL,
to Queues Central Bank India . initial parts are /joint
43 w
Fluid model framework
Fluid model
Fluid model where arrivals are points in a continuum [0, 1], server
serves at a fixed rate µ.
Server becomes active at time zero
Customers can come and queue up before or after time zero
Customer cost is additive and linear in waiting time and service
completion time (equivalently, in time spent in lounge)
Workshop on Congestion Gam
Sandeep Juneja, TIFR Visiting Professor
Strategic Arrivals
CAFRAL,
to Queues Central Bank India . initial parts are /joint
43 w
Fluid model framework
Fluid model
Fluid model where arrivals are points in a continuum [0, 1], server
serves at a fixed rate µ.
Server becomes active at time zero
Customers can come and queue up before or after time zero
Customer cost is additive and linear in waiting time and service
completion time (equivalently, in time spent in lounge)
Workshop on Congestion Gam
Sandeep Juneja, TIFR Visiting Professor
Strategic Arrivals
CAFRAL,
to Queues Central Bank India . initial parts are /joint
43 w
The arrival game is amongst customers in [0, 1].
Suppose that a customer s chooses her arrival time from distribution
Gs (·), s ∈ [0, 1]. The deterministic arrival profile is given by
Z
F (t) =
1
Gs (t)ds
0
for each t.
Workshop on Congestion Gam
Sandeep Juneja, TIFR Visiting Professor
Strategic Arrivals
CAFRAL,
to Queues Central Bank India . initial parts are /joint
43 w
The arrival game is amongst customers in [0, 1].
Suppose that a customer s chooses her arrival time from distribution
Gs (·), s ∈ [0, 1]. The deterministic arrival profile is given by
Z
F (t) =
1
Gs (t)ds
0
for each t.
Workshop on Congestion Gam
Sandeep Juneja, TIFR Visiting Professor
Strategic Arrivals
CAFRAL,
to Queues Central Bank India . initial parts are /joint
43 w
Customer Cost Function
Let WF (t) denote waiting time for arrival at time t when all
other customers have an arrival profile F .
Arrival at time t incurs cost
CF (t) = αWF (t) + β(t + WF (t))
More generally, the expected cost incurred by a customer s who
selects her arrival by sampling from probability distribution Gs is
Z ∞
CF (G ) =
(αWF (t) + β(t + WF (t))) dGs (t) .
−∞
Workshop on Congestion Gam
Sandeep Juneja, TIFR Visiting Professor
Strategic Arrivals
CAFRAL,
to Queues Central Bank India . initial parts are /joint
43 w
Customer Cost Function
Let WF (t) denote waiting time for arrival at time t when all
other customers have an arrival profile F .
Arrival at time t incurs cost
CF (t) = αWF (t) + β(t + WF (t))
More generally, the expected cost incurred by a customer s who
selects her arrival by sampling from probability distribution Gs is
Z ∞
CF (G ) =
(αWF (t) + β(t + WF (t))) dGs (t) .
−∞
Workshop on Congestion Gam
Sandeep Juneja, TIFR Visiting Professor
Strategic Arrivals
CAFRAL,
to Queues Central Bank India . initial parts are /joint
43 w
Customer Cost Function
Let WF (t) denote waiting time for arrival at time t when all
other customers have an arrival profile F .
Arrival at time t incurs cost
CF (t) = αWF (t) + β(t + WF (t))
More generally, the expected cost incurred by a customer s who
selects her arrival by sampling from probability distribution Gs is
Z ∞
CF (G ) =
(αWF (t) + β(t + WF (t))) dGs (t) .
−∞
Workshop on Congestion Gam
Sandeep Juneja, TIFR Visiting Professor
Strategic Arrivals
CAFRAL,
to Queues Central Bank India . initial parts are /joint
43 w
WF (t) is a function of QF (t) that depends on F
Queue length
QF (t) = F (t),
for t ≤ 0.
Waiting time
WF (t) = QF (t)/µ − t = F (t)/µ − t
When there is a jump at time t, replace F (t) by (F (t) + F (t − ))/2
Workshop on Congestion Gam
Sandeep Juneja, TIFR Visiting Professor
Strategic Arrivals
CAFRAL,
to Queues Central Bank India . initial parts are /joint
43 w
WF (t) is a function of QF (t) that depends on F
Queue length
QF (t) = F (t),
for t ≤ 0.
Waiting time
WF (t) = QF (t)/µ − t = F (t)/µ − t
When there is a jump at time t, replace F (t) by (F (t) + F (t − ))/2
Workshop on Congestion Gam
Sandeep Juneja, TIFR Visiting Professor
Strategic Arrivals
CAFRAL,
to Queues Central Bank India . initial parts are /joint
43 w
WF (t) is a function of QF (t) that depends on F
Queue length
QF (t) = F (t),
for t ≤ 0.
Waiting time
WF (t) = QF (t)/µ − t = F (t)/µ − t
When there is a jump at time t, replace F (t) by (F (t) + F (t − ))/2
Workshop on Congestion Gam
Sandeep Juneja, TIFR Visiting Professor
Strategic Arrivals
CAFRAL,
to Queues Central Bank India . initial parts are /joint
43 w
For t > 0, let t ∗ ∈ [0, t] denote the last time the queue was empty if
at all. Then,
QF (t) = F (t) − F (t ∗ ) − µ(t − t ∗ ).
and
WF (t) = QF (t)/µ.
Workshop on Congestion Gam
Sandeep Juneja, TIFR Visiting Professor
Strategic Arrivals
CAFRAL,
to Queues Central Bank India . initial parts are /joint
43 w
For t > 0, let t ∗ ∈ [0, t] denote the last time the queue was empty if
at all. Then,
QF (t) = F (t) − F (t ∗ ) − µ(t − t ∗ ).
and
WF (t) = QF (t)/µ.
Workshop on Congestion Gam
Sandeep Juneja, TIFR Visiting Professor
Strategic Arrivals
CAFRAL,
to Queues Central Bank India . initial parts are /joint
43 w
Evaluating WF (t), t > 0
Else, if queue was never empty
QF (t) = F (t) − µt.
and
WF (t) = QF (t)/µ = F (t)/µ − t,
Workshop on Congestion Gam
Sandeep Juneja, TIFR Visiting Professor
Strategic Arrivals
CAFRAL,
to Queues Central Bank India . initial parts are /joint
43 w
Nash Equilibrium
A multi-strategy {Gs (·), s ∈ [0, 1]} with arrival profile
R1
F (t) = 0 Gs (t)ds for each t is a Nash equilibrium point if
For any customer s ∈ [0, 1],
CF (Gs ) ≤ CF (G̃ ),
for every CDF G̃ .
That is, no customer s can improve her cost by modifying her
own arrival time distribution.
This is equivalent to an existence of an arrival profile F
such that CF (t) is constant on support of F and at least
as much elsewhere.
Workshop on Congestion Gam
Sandeep Juneja, TIFR Visiting Professor
Strategic Arrivals
CAFRAL,
to Queues Central Bank India . initial parts are /joint
43 w
Nash Equilibrium
A multi-strategy {Gs (·), s ∈ [0, 1]} with arrival profile
R1
F (t) = 0 Gs (t)ds for each t is a Nash equilibrium point if
For any customer s ∈ [0, 1],
CF (Gs ) ≤ CF (G̃ ),
for every CDF G̃ .
That is, no customer s can improve her cost by modifying her
own arrival time distribution.
This is equivalent to an existence of an arrival profile F
such that CF (t) is constant on support of F and at least
as much elsewhere.
Workshop on Congestion Gam
Sandeep Juneja, TIFR Visiting Professor
Strategic Arrivals
CAFRAL,
to Queues Central Bank India . initial parts are /joint
43 w
Equilibrium profile implies equilibrium multi-strategy
Workshop on Congestion Gam
Sandeep Juneja, TIFR Visiting Professor
Strategic Arrivals
CAFRAL,
to Queues Central Bank India . initial parts are /joint
43 w
Non-equilibrium profile implies non-equilibrium
multi-strategy
Workshop on Congestion Gam
Sandeep Juneja, TIFR Visiting Professor
Strategic Arrivals
CAFRAL,
to Queues Central Bank India . initial parts are /joint
43 w
Hunt for Equilibrium Profile
Workshop on Congestion Gam
Sandeep Juneja, TIFR Visiting Professor
Strategic Arrivals
CAFRAL,
to Queues Central Bank India . initial parts are /joint
43 w
Useful Insights
Let t ∗ = inf{t ≥ 0 : F (t) < µt}.
In Nash Equilibrium, first time the server has spare capacity is
when all customers are served. That is t ∗ = 1/µ.
Similarly, in Nash equilibrium there can be no point masses in F .
Workshop on Congestion Gam
Sandeep Juneja, TIFR Visiting Professor
Strategic Arrivals
CAFRAL,
to Queues Central Bank India . initial parts are /joint
43 w
Useful Insights
Let t ∗ = inf{t ≥ 0 : F (t) < µt}.
In Nash Equilibrium, first time the server has spare capacity is
when all customers are served. That is t ∗ = 1/µ.
Similarly, in Nash equilibrium there can be no point masses in F .
Workshop on Congestion Gam
Sandeep Juneja, TIFR Visiting Professor
Strategic Arrivals
CAFRAL,
to Queues Central Bank India . initial parts are /joint
43 w
Cost Function under Equilibrium
If F (·) denotes an equilibrium profile Then, WF (t) equals
F (t)/µ − t,
for t ≤ 1/µ.
The cost function CF (t) equals
β(t + WF (t)) + αWF (t) = (α + β)F (t)/µ − αt
Workshop on Congestion Gam
Sandeep Juneja, TIFR Visiting Professor
Strategic Arrivals
CAFRAL,
to Queues Central Bank India . initial parts are /joint
43 w
Cost Function under Equilibrium
If F (·) denotes an equilibrium profile Then, WF (t) equals
F (t)/µ − t,
for t ≤ 1/µ.
The cost function CF (t) equals
β(t + WF (t)) + αWF (t) = (α + β)F (t)/µ − αt
Workshop on Congestion Gam
Sandeep Juneja, TIFR Visiting Professor
Strategic Arrivals
CAFRAL,
to Queues Central Bank India . initial parts are /joint
43 w
Cost Function under Equilibrium
CF (t) = (α + β)F (t)/µ − αt
Last arrival at time ≤ 1/µ. This must equal 1/µ to improve cost.
Hence, F (1/µ) = 1, and CF (t) = β/µ on support of F .
Furthermore
F (t) =
(t − t0 )
,
(t1 − t0 )
where t1 = 1/µ and t0 = −β/(αµ). F (t) = 0 for t < t0 and
F (t) = 1 for t > t1 .
Workshop on Congestion Gam
Sandeep Juneja, TIFR Visiting Professor
Strategic Arrivals
CAFRAL,
to Queues Central Bank India . initial parts are /joint
43 w
Cost Function under Equilibrium
CF (t) = (α + β)F (t)/µ − αt
Last arrival at time ≤ 1/µ. This must equal 1/µ to improve cost.
Hence, F (1/µ) = 1, and CF (t) = β/µ on support of F .
Furthermore
F (t) =
(t − t0 )
,
(t1 − t0 )
where t1 = 1/µ and t0 = −β/(αµ). F (t) = 0 for t < t0 and
F (t) = 1 for t > t1 .
Workshop on Congestion Gam
Sandeep Juneja, TIFR Visiting Professor
Strategic Arrivals
CAFRAL,
to Queues Central Bank India . initial parts are /joint
43 w
Cost Function under Equilibrium
CF (t) = (α + β)F (t)/µ − αt
Last arrival at time ≤ 1/µ. This must equal 1/µ to improve cost.
Hence, F (1/µ) = 1, and CF (t) = β/µ on support of F .
Furthermore
F (t) =
(t − t0 )
,
(t1 − t0 )
where t1 = 1/µ and t0 = −β/(αµ). F (t) = 0 for t < t0 and
F (t) = 1 for t > t1 .
Workshop on Congestion Gam
Sandeep Juneja, TIFR Visiting Professor
Strategic Arrivals
CAFRAL,
to Queues Central Bank India . initial parts are /joint
43 w
Queue Length Process
0
F (t) = µ
α
α+β
for t0 < t < t1 = 1/µ.
Workshop on Congestion Gam
Sandeep Juneja, TIFR Visiting Professor
Strategic Arrivals
CAFRAL,
to Queues Central Bank India . initial parts are /joint
43 w
Socially Optimal Solution
The smallest value of waiting time is zero.
Total time to service is minimized if the server serves at the
fastest possible rate. So the average service time is 1/(2µ) and
the overall cost is β/(2µ)
Price of anarchy equals two
Workshop on Congestion Gam
Sandeep Juneja, TIFR Visiting Professor
Strategic Arrivals
CAFRAL,
to Queues Central Bank India . initial parts are /joint
43 w
Socially Optimal Solution
The smallest value of waiting time is zero.
Total time to service is minimized if the server serves at the
fastest possible rate. So the average service time is 1/(2µ) and
the overall cost is β/(2µ)
Price of anarchy equals two
Workshop on Congestion Gam
Sandeep Juneja, TIFR Visiting Professor
Strategic Arrivals
CAFRAL,
to Queues Central Bank India . initial parts are /joint
43 w
Socially Optimal Solution
The smallest value of waiting time is zero.
Total time to service is minimized if the server serves at the
fastest possible rate. So the average service time is 1/(2µ) and
the overall cost is β/(2µ)
Price of anarchy equals two
Workshop on Congestion Gam
Sandeep Juneja, TIFR Visiting Professor
Strategic Arrivals
CAFRAL,
to Queues Central Bank India . initial parts are /joint
43 w
Finite Population Analysis
Workshop on Congestion Gam
Sandeep Juneja, TIFR Visiting Professor
Strategic Arrivals
CAFRAL,
to Queues Central Bank India . initial parts are /joint
43 w
Finite Population Analysis
N customers, each requires an exponentially distributed service.
F = {Fi : i ≤ N} denotes a collection of arrival time
distributions of all customers.
The expected cost
Ci (F) = EF (αWi + βTi ). ,
An arrival profile {Fi , i = 1, . . . , N} is a Nash equilibrium point
if
Ci (Fi , F −i ) ≤ Ci (F˜i , F −i )
for every customer i and every CDF F˜i .
Workshop on Congestion Gam
Sandeep Juneja, TIFR Visiting Professor
Strategic Arrivals
CAFRAL,
to Queues Central Bank India . initial parts are /joint
43 w
Finite Population Analysis
N customers, each requires an exponentially distributed service.
F = {Fi : i ≤ N} denotes a collection of arrival time
distributions of all customers.
The expected cost
Ci (F) = EF (αWi + βTi ). ,
An arrival profile {Fi , i = 1, . . . , N} is a Nash equilibrium point
if
Ci (Fi , F −i ) ≤ Ci (F˜i , F −i )
for every customer i and every CDF F˜i .
Workshop on Congestion Gam
Sandeep Juneja, TIFR Visiting Professor
Strategic Arrivals
CAFRAL,
to Queues Central Bank India . initial parts are /joint
43 w
Finite Population Analysis
N customers, each requires an exponentially distributed service.
F = {Fi : i ≤ N} denotes a collection of arrival time
distributions of all customers.
The expected cost
Ci (F) = EF (αWi + βTi ). ,
An arrival profile {Fi , i = 1, . . . , N} is a Nash equilibrium point
if
Ci (Fi , F −i ) ≤ Ci (F˜i , F −i )
for every customer i and every CDF F˜i .
Workshop on Congestion Gam
Sandeep Juneja, TIFR Visiting Professor
Strategic Arrivals
CAFRAL,
to Queues Central Bank India . initial parts are /joint
43 w
Finite Population Analysis
N customers, each requires an exponentially distributed service.
F = {Fi : i ≤ N} denotes a collection of arrival time
distributions of all customers.
The expected cost
Ci (F) = EF (αWi + βTi ). ,
An arrival profile {Fi , i = 1, . . . , N} is a Nash equilibrium point
if
Ci (Fi , F −i ) ≤ Ci (F˜i , F −i )
for every customer i and every CDF F˜i .
Workshop on Congestion Gam
Sandeep Juneja, TIFR Visiting Professor
Strategic Arrivals
CAFRAL,
to Queues Central Bank India . initial parts are /joint
43 w
Alternative definition of Nash Equilibrium
An arrival profile {Fi , i = 1, . . . , N} is a Nash equilibrium point
if, and only if, for every i there exists a constant ci so that
(i) Ci (t, F −i ) ≥ ci for all t.
(ii) Ci (t, F −i ) = ci on a set of Fi -measure 1.
Workshop on Congestion Gam
Sandeep Juneja, TIFR Visiting Professor
Strategic Arrivals
CAFRAL,
to Queues Central Bank India . initial parts are /joint
43 w
Nash Equilibrium must be symmetric
Suppose ci < cj . Then customer j can improve cost by coming
just before the earliest time in support of Fi .
If the two df’s Fi and Fj differ, e.g. Fi (t) = Fj (t) for t ≤ t ∗ and
then Fj (t) > Fi (t) over an interval just after t ∗ .
This implies that Q −i (t) > Q −j (t) for some t, implying that the costs
incurred by i and j are different at t.
Workshop on Congestion Gam
Sandeep Juneja, TIFR Visiting Professor
Strategic Arrivals
CAFRAL,
to Queues Central Bank India . initial parts are /joint
43 w
Nash Equilibrium must be symmetric
Suppose ci < cj . Then customer j can improve cost by coming
just before the earliest time in support of Fi .
If the two df’s Fi and Fj differ, e.g. Fi (t) = Fj (t) for t ≤ t ∗ and
then Fj (t) > Fi (t) over an interval just after t ∗ .
This implies that Q −i (t) > Q −j (t) for some t, implying that the costs
incurred by i and j are different at t.
Workshop on Congestion Gam
Sandeep Juneja, TIFR Visiting Professor
Strategic Arrivals
CAFRAL,
to Queues Central Bank India . initial parts are /joint
43 w
Nash Equilibrium Functional ODE
Setting the cost function derivative with respect to time to zero
α
N −1 0
F (t) =
− P0 (t),
µ
α+β
t ∈ [0, tb ). and
N −1 0
α
F (t) =
,
µ
α+β
for t ∈ (ta , 0).
Workshop on Congestion Gam
Sandeep Juneja, TIFR Visiting Professor
Strategic Arrivals
CAFRAL,
to Queues Central Bank India . initial parts are /joint
43 w
Nash Equilibrium Functional ODE
Setting the cost function derivative with respect to time to zero
α
N −1 0
F (t) =
− P0 (t),
µ
α+β
t ∈ [0, tb ). and
N −1 0
α
F (t) =
,
µ
α+β
for t ∈ (ta , 0).
Workshop on Congestion Gam
Sandeep Juneja, TIFR Visiting Professor
Strategic Arrivals
CAFRAL,
to Queues Central Bank India . initial parts are /joint
43 w
Computation Results
0.7
M=1
M=2
M=4
M=16
M=64
Fluid
0.6
0.5
G’(t)
0.4
0.3
0.2
0.1
0
−1
−0.5
0
0.5
1
1.5
Workshop on Congestion Gam
t
Sandeep Juneja, TIFR Visiting Professor
Strategic Arrivals
CAFRAL,
to Queues Central Bank India . initial parts are /joint
43 w
Convergence to the fluid limit
Theorem
√ N ), in
F (Nt) converges to a uniform distribution on [− µ1 αβ , µ1 ] at rate o( log
N
the sense that
N −1 0
α
F (t) =
− P0 (t)1{t ≥ 0} ,
µ
α+β
t ∈ (ta , tb )
is satisfied with
1β
log N
1β
ta
=−
− o( √ ) < −
N −1
µα
µα
N
tb
1
log N
= + o( √ )
N −1
µ
N
and
P0 (t) ≤
t
1
log N
1
for
≤ − o( √ ) .
N
N
µ
N
(1)
(2)
(3)
Workshop on Congestion Gam
Sandeep Juneja, TIFR Visiting Professor
Strategic Arrivals
CAFRAL,
to Queues Central Bank India . initial parts are /joint
43 w
Proof idea: Inductively show that P0 (t) is small
Workshop on Congestion Gam
Sandeep Juneja, TIFR Visiting Professor
Strategic Arrivals
CAFRAL,
to Queues Central Bank India . initial parts are /joint
43 w
Torestinloungeorwaitinqueue
Concertqueue,queuesizeknown
Workshop on Congestion Gam
Sandeep Juneja, TIFR Visiting Professor
Strategic Arrivals
CAFRAL,
to Queues Central Bank India . initial parts are /joint
43 w
Finite Population, can observe the queue at a ‘Concert’
Finite population of size N.
Customers start by all being in the lounge. They can observe the lounge as
well as the queue size.
Queue service starts at time zero. Exponential service times.
Lounge unit time cost β < α unit wait cost.
We assume that if many people in lounge wish to join the queue, only
one succeeds at random.
Workshop on Congestion Gam
Sandeep Juneja, TIFR Visiting Professor
Strategic Arrivals
CAFRAL,
to Queues Central Bank India . initial parts are /joint
43 w
Finite Population, can observe the queue at a ‘Concert’
Finite population of size N.
Customers start by all being in the lounge. They can observe the lounge as
well as the queue size.
Queue service starts at time zero. Exponential service times.
Lounge unit time cost β < α unit wait cost.
We assume that if many people in lounge wish to join the queue, only
one succeeds at random.
Workshop on Congestion Gam
Sandeep Juneja, TIFR Visiting Professor
Strategic Arrivals
CAFRAL,
to Queues Central Bank India . initial parts are /joint
43 w
Finite Population, can observe the queue at a ‘Concert’
Finite population of size N.
Customers start by all being in the lounge. They can observe the lounge as
well as the queue size.
Queue service starts at time zero. Exponential service times.
Lounge unit time cost β < α unit wait cost.
We assume that if many people in lounge wish to join the queue, only
one succeeds at random.
Workshop on Congestion Gam
Sandeep Juneja, TIFR Visiting Professor
Strategic Arrivals
CAFRAL,
to Queues Central Bank India . initial parts are /joint
43 w
Finite Population, can observe the queue at a ‘Concert’
Finite population of size N.
Customers start by all being in the lounge. They can observe the lounge as
well as the queue size.
Queue service starts at time zero. Exponential service times.
Lounge unit time cost β < α unit wait cost.
We assume that if many people in lounge wish to join the queue, only
one succeeds at random.
Workshop on Congestion Gam
Sandeep Juneja, TIFR Visiting Professor
Strategic Arrivals
CAFRAL,
to Queues Central Bank India . initial parts are /joint
43 w
Finite Population, can observe the queue at a ‘Concert’
Finite population of size N.
Customers start by all being in the lounge. They can observe the lounge as
well as the queue size.
Queue service starts at time zero. Exponential service times.
Lounge unit time cost β < α unit wait cost.
We assume that if many people in lounge wish to join the queue, only
one succeeds at random.
Workshop on Congestion Gam
Sandeep Juneja, TIFR Visiting Professor
Strategic Arrivals
CAFRAL,
to Queues Central Bank India . initial parts are /joint
43 w
Equilbrium policy has the threshold {m(n)} form.
If n people in lounge, then for queue m ≤ m(n), people in lounge
compete to join the queue.
For m > m(n), people in the lounge choose to wait.
Workshop on Congestion Gam
Sandeep Juneja, TIFR Visiting Professor
Strategic Arrivals
CAFRAL,
to Queues Central Bank India . initial parts are /joint
43 w
Equilbrium policy has the threshold {m(n)} form.
If n people in lounge, then for queue m ≤ m(n), people in lounge
compete to join the queue.
For m > m(n), people in the lounge choose to wait.
Workshop on Congestion Gam
Sandeep Juneja, TIFR Visiting Professor
Strategic Arrivals
CAFRAL,
to Queues Central Bank India . initial parts are /joint
43 w
Illustrative threshold equilibrium policy
m(n)
5
6
3
n=3
n=4
Workshop on Congestion Gam
Sandeep Juneja, TIFR Visiting Professor
Strategic Arrivals
CAFRAL,
to Queues Central Bank India . initial parts are /joint
43 w
n=2
The related equations are simple
For m ≤ m(n),
C (m, n) =
1
n−1
αm/µ +
C (m + 1, n − 1) ≤ C (m + 1, n − 1)
n
n
For m > m(n),
C (m, n) = β/µ + C (m − 1, n) ≤ αm/µ.
Workshop on Congestion Gam
Sandeep Juneja, TIFR Visiting Professor
Strategic Arrivals
CAFRAL,
to Queues Central Bank India . initial parts are /joint
43 w
The related equations are simple
For m ≤ m(n),
C (m, n) =
1
n−1
αm/µ +
C (m + 1, n − 1) ≤ C (m + 1, n − 1)
n
n
For m > m(n),
C (m, n) = β/µ + C (m − 1, n) ≤ αm/µ.
Workshop on Congestion Gam
Sandeep Juneja, TIFR Visiting Professor
Strategic Arrivals
CAFRAL,
to Queues Central Bank India . initial parts are /joint
43 w
These are easily solved
C (m, 1) = mβ/µ
m(n) = max{m : αm/µ ≤ C (m + 1, n − 1)}
For m ≤ m(n),
C (m, n) =
1
n−1
αm/µ +
C (m + 1, n − 1).
n
n
For m > m(n),
C (m, n) = β/µ + C (m − 1, n).
Workshop on Congestion Gam
Sandeep Juneja, TIFR Visiting Professor
Strategic Arrivals
CAFRAL,
to Queues Central Bank India . initial parts are /joint
43 w
These are easily solved
C (m, 1) = mβ/µ
m(n) = max{m : αm/µ ≤ C (m + 1, n − 1)}
For m ≤ m(n),
C (m, n) =
1
n−1
αm/µ +
C (m + 1, n − 1).
n
n
For m > m(n),
C (m, n) = β/µ + C (m − 1, n).
Workshop on Congestion Gam
Sandeep Juneja, TIFR Visiting Professor
Strategic Arrivals
CAFRAL,
to Queues Central Bank India . initial parts are /joint
43 w
These are easily solved
C (m, 1) = mβ/µ
m(n) = max{m : αm/µ ≤ C (m + 1, n − 1)}
For m ≤ m(n),
C (m, n) =
1
n−1
αm/µ +
C (m + 1, n − 1).
n
n
For m > m(n),
C (m, n) = β/µ + C (m − 1, n).
Workshop on Congestion Gam
Sandeep Juneja, TIFR Visiting Professor
Strategic Arrivals
CAFRAL,
to Queues Central Bank India . initial parts are /joint
43 w
These are easily solved
C (m, 1) = mβ/µ
m(n) = max{m : αm/µ ≤ C (m + 1, n − 1)}
For m ≤ m(n),
C (m, n) =
1
n−1
αm/µ +
C (m + 1, n − 1).
n
n
For m > m(n),
C (m, n) = β/µ + C (m − 1, n).
Workshop on Congestion Gam
Sandeep Juneja, TIFR Visiting Professor
Strategic Arrivals
CAFRAL,
to Queues Central Bank India . initial parts are /joint
43 w
Some results
β
m(n)
→
n
α−β
C (m(n) + 1, n)
αβ
→
n
α−β
Strategy {m(n)} is a weakly dominant strategy when the server is on.
Thus, unique Nash equilibrium.
Price of anarchy is bounded from above by 2.
Workshop on Congestion Gam
Sandeep Juneja, TIFR Visiting Professor
Strategic Arrivals
CAFRAL,
to Queues Central Bank India . initial parts are /joint
43 w
Some results
β
m(n)
→
n
α−β
C (m(n) + 1, n)
αβ
→
n
α−β
Strategy {m(n)} is a weakly dominant strategy when the server is on.
Thus, unique Nash equilibrium.
Price of anarchy is bounded from above by 2.
Workshop on Congestion Gam
Sandeep Juneja, TIFR Visiting Professor
Strategic Arrivals
CAFRAL,
to Queues Central Bank India . initial parts are /joint
43 w
Some results
β
m(n)
→
n
α−β
C (m(n) + 1, n)
αβ
→
n
α−β
Strategy {m(n)} is a weakly dominant strategy when the server is on.
Thus, unique Nash equilibrium.
Price of anarchy is bounded from above by 2.
Workshop on Congestion Gam
Sandeep Juneja, TIFR Visiting Professor
Strategic Arrivals
CAFRAL,
to Queues Central Bank India . initial parts are /joint
43 w
Some results
β
m(n)
→
n
α−β
C (m(n) + 1, n)
αβ
→
n
α−β
Strategy {m(n)} is a weakly dominant strategy when the server is on.
Thus, unique Nash equilibrium.
Price of anarchy is bounded from above by 2.
Workshop on Congestion Gam
Sandeep Juneja, TIFR Visiting Professor
Strategic Arrivals
CAFRAL,
to Queues Central Bank India . initial parts are /joint
43 w
Torestinloungeorwaitinqueue
M/M/1queue,queuesizeknown
Workshop on Congestion Gam
Sandeep Juneja, TIFR Visiting Professor
Strategic Arrivals
CAFRAL,
to Queues Central Bank India . initial parts are /joint
43 w
M/M/1 queue with lounge facility
Need to get service arises as a Poisson process.
Arrival joins lounge. Can see others in the lounge.
For
µ
α
≤
β
µ−λ
dominant strategy to join the queue.
Lounge remains empty. Everyone joins the queue.
Price of anarchy equals
α
β.
Using Little’s Law.
Workshop on Congestion Gam
Sandeep Juneja, TIFR Visiting Professor
Strategic Arrivals
CAFRAL,
to Queues Central Bank India . initial parts are /joint
43 w
M/M/1 queue with lounge facility
Need to get service arises as a Poisson process.
Arrival joins lounge. Can see others in the lounge.
For
µ
α
≤
β
µ−λ
dominant strategy to join the queue.
Lounge remains empty. Everyone joins the queue.
Price of anarchy equals
α
β.
Using Little’s Law.
Workshop on Congestion Gam
Sandeep Juneja, TIFR Visiting Professor
Strategic Arrivals
CAFRAL,
to Queues Central Bank India . initial parts are /joint
43 w
M/M/1 queue with lounge facility
Need to get service arises as a Poisson process.
Arrival joins lounge. Can see others in the lounge.
For
µ
α
≤
β
µ−λ
dominant strategy to join the queue.
Lounge remains empty. Everyone joins the queue.
Price of anarchy equals
α
β.
Using Little’s Law.
Workshop on Congestion Gam
Sandeep Juneja, TIFR Visiting Professor
Strategic Arrivals
CAFRAL,
to Queues Central Bank India . initial parts are /joint
43 w
M/M/1 queue with lounge facility
Need to get service arises as a Poisson process.
Arrival joins lounge. Can see others in the lounge.
For
µ
α
≤
β
µ−λ
dominant strategy to join the queue.
Lounge remains empty. Everyone joins the queue.
Price of anarchy equals
α
β.
Using Little’s Law.
Workshop on Congestion Gam
Sandeep Juneja, TIFR Visiting Professor
Strategic Arrivals
CAFRAL,
to Queues Central Bank India . initial parts are /joint
43 w
M/M/1 queue with lounge facility
For
µ
α
>
β
µ−λ
Equilbrium policy has the threshold {m(n)} form.
If n people in lounge, then for queue m ≤ m(n), people in lounge
compete to join the queue.
Equations similar to before. Price of anarchy is bounded from above
by
µ
.
µ−λ
Workshop on Congestion Gam
Sandeep Juneja, TIFR Visiting Professor
Strategic Arrivals
CAFRAL,
to Queues Central Bank India . initial parts are /joint
43 w
M/M/1 queue with lounge facility
For
µ
α
>
β
µ−λ
Equilbrium policy has the threshold {m(n)} form.
If n people in lounge, then for queue m ≤ m(n), people in lounge
compete to join the queue.
Equations similar to before. Price of anarchy is bounded from above
by
µ
.
µ−λ
Workshop on Congestion Gam
Sandeep Juneja, TIFR Visiting Professor
Strategic Arrivals
CAFRAL,
to Queues Central Bank India . initial parts are /joint
43 w
M/M/1 queue with lounge facility
For
µ
α
>
β
µ−λ
Equilbrium policy has the threshold {m(n)} form.
If n people in lounge, then for queue m ≤ m(n), people in lounge
compete to join the queue.
Equations similar to before. Price of anarchy is bounded from above
by
µ
.
µ−λ
Workshop on Congestion Gam
Sandeep Juneja, TIFR Visiting Professor
Strategic Arrivals
CAFRAL,
to Queues Central Bank India . initial parts are /joint
43 w
M/M/1 queue with lounge facility
For
µ
α
>
β
µ−λ
Equilbrium policy has the threshold {m(n)} form.
If n people in lounge, then for queue m ≤ m(n), people in lounge
compete to join the queue.
Equations similar to before. Price of anarchy is bounded from above
by
µ
.
µ−λ
Workshop on Congestion Gam
Sandeep Juneja, TIFR Visiting Professor
Strategic Arrivals
CAFRAL,
to Queues Central Bank India . initial parts are /joint
43 w
Limiting Queue Buffer to Control PoA
For
µ
α
≤
β
µ−λ
by limiting queue size, customers forced to remain in lounge.
Can numerically see that lounge may have customers in this case!
Can bound PoA by
α
(1 − ρk+1 ) + ρk+1 ,
β
where k is the queue buffer.
Workshop on Congestion Gam
Sandeep Juneja, TIFR Visiting Professor
Strategic Arrivals
CAFRAL,
to Queues Central Bank India . initial parts are /joint
43 w
Limiting Queue Buffer to Control PoA
For
µ
α
≤
β
µ−λ
by limiting queue size, customers forced to remain in lounge.
Can numerically see that lounge may have customers in this case!
Can bound PoA by
α
(1 − ρk+1 ) + ρk+1 ,
β
where k is the queue buffer.
Workshop on Congestion Gam
Sandeep Juneja, TIFR Visiting Professor
Strategic Arrivals
CAFRAL,
to Queues Central Bank India . initial parts are /joint
43 w
Limiting Queue Buffer to Control PoA
For
µ
α
≤
β
µ−λ
by limiting queue size, customers forced to remain in lounge.
Can numerically see that lounge may have customers in this case!
Can bound PoA by
α
(1 − ρk+1 ) + ρk+1 ,
β
where k is the queue buffer.
Workshop on Congestion Gam
Sandeep Juneja, TIFR Visiting Professor
Strategic Arrivals
CAFRAL,
to Queues Central Bank India . initial parts are /joint
43 w
Conclusion
Discussed strategic arrivals in the concert queue setting when the
queue is not observed by customers; customers are non-atomic as
well as when they are atomic.
Strategic arrivals in concert queue setting when customers in
lounge can observe the queue.
Considered lounge in M/M/1 setting where we do steady state
analysis of strategic queue joining behaviour.
Workshop on Congestion Gam
Sandeep Juneja, TIFR Visiting Professor
Strategic Arrivals
CAFRAL,
to Queues Central Bank India . initial parts are /joint
43 w
Conclusion
Discussed strategic arrivals in the concert queue setting when the
queue is not observed by customers; customers are non-atomic as
well as when they are atomic.
Strategic arrivals in concert queue setting when customers in
lounge can observe the queue.
Considered lounge in M/M/1 setting where we do steady state
analysis of strategic queue joining behaviour.
Workshop on Congestion Gam
Sandeep Juneja, TIFR Visiting Professor
Strategic Arrivals
CAFRAL,
to Queues Central Bank India . initial parts are /joint
43 w
Conclusion
Discussed strategic arrivals in the concert queue setting when the
queue is not observed by customers; customers are non-atomic as
well as when they are atomic.
Strategic arrivals in concert queue setting when customers in
lounge can observe the queue.
Considered lounge in M/M/1 setting where we do steady state
analysis of strategic queue joining behaviour.
Workshop on Congestion Gam
Sandeep Juneja, TIFR Visiting Professor
Strategic Arrivals
CAFRAL,
to Queues Central Bank India . initial parts are /joint
43 w