Coupling and capacity value in an M/M/1/C
queue
Peter Braunsteins
Supervisors: Professor Peter Taylor and Dr Sophie Hautphenne
Department of Mathematics and Statistics, The University of Melbourne
8th April 2015
Peter Braunsteins
Coupling and capacity value in an M/M/1/C queue
Department of Mathematics and Statistics, The University of Melbourne
M/M/1/C Loss System
I
Consider an M/M/1/C queue run by a manager who has the option
of buying or selling capacity at the start of each time period
I
How much should the manager pay for an additional unit of
capacity?
I
What price should the manager accept to sell a unit of capacity?
I
Chiera and Taylor (2002) calculated these prices for an M/M/C /C
system
Peter Braunsteins
Coupling and capacity value in an M/M/1/C queue
Department of Mathematics and Statistics, The University of Melbourne
M/M/1/C Loss System
I
There’s a finite planning horizon of length T
I
Each customer served generates θ dollars
I
There are initially n customers in the queue
I
Customers arrive to the queue according to the cumulative arrival
process; {A(t) : 0 ≤ t ≤ T }
I
Customers are served according to the potential service process;
{S(t) : 0 ≤ t ≤ T }
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These are independent Poisson processes with rates λ and µ,
respectively.
Peter Braunsteins
Coupling and capacity value in an M/M/1/C queue
Department of Mathematics and Statistics, The University of Melbourne
Expected Lost Revenue
I
We say that each customer rejected from the queue is a lost
opportunity which costs θ dollars
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We compute Rn,C (t), the expected revenue lost in the time interval
[0, t] given an M/M/1/C system with capacity C and initial queue
length n ∈ {0, 1, ..., C }
Peter Braunsteins
Coupling and capacity value in an M/M/1/C queue
Department of Mathematics and Statistics, The University of Melbourne
Queue Length Process
I
Let Qn,C (t) represent the number of customers in the queue at time
t given an initial queue length n and capacity C .
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Qn,C (t) = n + [A(t) − Un,C (t)] − [S(t) − Ln,C (t)], where
I
I
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Ln,C (t) counts the number of unrealized potential services
Un,C (t) counts the number of customers lost
Rn,C (t) = θE [Un,C (t)]
Peter Braunsteins
Coupling and capacity value in an M/M/1/C queue
Department of Mathematics and Statistics, The University of Melbourne
Queue Length Process
Figure 1: Possible realization of the queue length process, Q0,4 (t). Points at
queue length C represent lost customers and points at queue length 0 represent
unrealized services. In this example 8 customers are lost and 5 services are
unrealized.
Peter Braunsteins
Coupling and capacity value in an M/M/1/C queue
Department of Mathematics and Statistics, The University of Melbourne
Expected Lost Revenue - n = {0, 1, ..., C }
Figure 2: Expected lost revenue function for n = {0, 1, ..., C } when C = 4,
θ = 1, λ = 3 and µ = 5 (low blocking system).
Peter Braunsteins
Coupling and capacity value in an M/M/1/C queue
Department of Mathematics and Statistics, The University of Melbourne
Expected Lost Revenue - n = {0, 1, ..., C }
Figure 3: Expected lost revenue function for n = {0, 1, ..., C } when C = 4,
θ = 1, λ = 5 and µ = 3 (high blocking system).
Peter Braunsteins
Coupling and capacity value in an M/M/1/C queue
Department of Mathematics and Statistics, The University of Melbourne
Buying and Selling Prices
Let the buying price Bn,C (t), be,
I
Bn,C (t) = Rn,C (t) − Rn,C +1 (t), for all n ∈ {0, ..., C }
I
Let the selling price Sn,C (t) be,
Rn,C −1 (t) − Rn,C (t),
n ∈ {0, ..., C − 1}
Sn,C (t) =
RC −1,C −1 (t) − RC ,C (t) + θ, n = C .
Peter Braunsteins
Coupling and capacity value in an M/M/1/C queue
Department of Mathematics and Statistics, The University of Melbourne
Buying and Selling Prices
Figure 4: Buying (solid) and selling (dotted) price function for n = 3, 4, 5 when
C = 5, λ = 3 and µ = 5 (low blocking system)
Peter Braunsteins
Coupling and capacity value in an M/M/1/C queue
Department of Mathematics and Statistics, The University of Melbourne
Buying and Selling Prices
Figure 5: Buying (solid) and selling (dotted) price function for n = 3, 4, 5 when
C = 5, λ = 5 and µ = 3 (high blocking system)
Peter Braunsteins
Coupling and capacity value in an M/M/1/C queue
Department of Mathematics and Statistics, The University of Melbourne
Buying and Selling Prices
Theorem
Consider two M/M/1/C queues with identical values of C and θ, a ‘high
blocking’ queue with parameters λHB , µHB and a ‘low blocking’ queue
with parameters λLB , µLB , which are such that λHB > λLB , λHB = µLB
HB
and µHB = λLB . If Sn,C
(t) is the selling price of capacity in the high
LB
blocking queue and Sn,C (t) is the selling price of capacity in the low
blocking queue, then
LB
SCHB
,C (t) = SC ,C (t) for all t ≥ 0.
Peter Braunsteins
Coupling and capacity value in an M/M/1/C queue
Department of Mathematics and Statistics, The University of Melbourne
Coupling
I
The buying and selling prices are expressed as the difference in
expected lost revenue from two queueing systems
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We transition to a probability space where both queueing systems
are generated by the same arrival and potential service processes
Sn,C (t) = Rn,C −1 (t) − Rn,C (t)
= E(Un,C −1 (t)) − E(Un,C (t))
= E Uˆn,C −1 (t) − Uˆn,C (t)
Peter Braunsteins
Coupling and capacity value in an M/M/1/C queue
Department of Mathematics and Statistics, The University of Melbourne
Coupling
Figure 6: Possible realization of the queue length processes Qˆ1,3 (t) (blue
dashed) and Qˆ1,2 (t) (red solid). Points at queue length C represent lost
customers and points at queue length 0 represent unrealized services.
Peter Braunsteins
Coupling and capacity value in an M/M/1/C queue
Department of Mathematics and Statistics, The University of Melbourne
Coupling
Figure 7: Possible realization of the queue length processes Qˆ1,3 (t) (blue
dashed) and Qˆ1,2 (t) (red solid). Points at queue length C represent lost
customers and points at queue length 0 represent unrealized services.
Peter Braunsteins
Coupling and capacity value in an M/M/1/C queue
Department of Mathematics and Statistics, The University of Melbourne
Coupling
Figure 8: Possible realization of the queue length processes Qˆ1,3 (t) (blue
dashed) and Qˆ1,2 (t) (red solid). Points at queue length C represent lost
customers and points at queue length 0 represent unrealized services.
Peter Braunsteins
Coupling and capacity value in an M/M/1/C queue
Department of Mathematics and Statistics, The University of Melbourne
Coupling
Figure 9: Possible realization of the queue length processes Qˆ1,3 (t) (blue
dashed) and Qˆ1,2 (t) (red solid). Points at queue length C represent lost
customers and points at queue length 0 represent unrealized services.
Peter Braunsteins
Coupling and capacity value in an M/M/1/C queue
Department of Mathematics and Statistics, The University of Melbourne
Coupling
Figure 10: Possible realization of the queue length processes Qˆ1,3 (t) (blue
dashed) and Qˆ1,2 (t) (red solid). Points at queue length C represent lost
customers and points at queue length 0 represent unrealized services.
Peter Braunsteins
Coupling and capacity value in an M/M/1/C queue
Department of Mathematics and Statistics, The University of Melbourne
Coupling
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{Uˆn,C −1 (t) − Uˆn,C (t)} is a delayed renewal process
I
HB
LB
HB
LB
Since Ti,U
= Ti,L
and Ti,L
= Ti,U
, the inter-renewal time,
Ti,L + Ti,U , has the same distribution in the high blocking and low
blocking cases
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The delay, T1 , is different in the high blocking and low blocking
cases except when n = C , in which case T1 = 0.
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Thus,
d
d
ˆHB
ˆHB
SCHB
,C (t) = θE 1 + UC −1,C −1 (t) − UC ,C (t)
= θE 1 + UˆCLB−1,C −1 (t) − UˆCLB,C (t)
= SCLB,C (t)
Peter Braunsteins
Coupling and capacity value in an M/M/1/C queue
Department of Mathematics and Statistics, The University of Melbourne
Questions?
Peter Braunsteins
Coupling and capacity value in an M/M/1/C queue
Department of Mathematics and Statistics, The University of Melbourne