EE 685 proposals
Network utility optimal admission and scheduling
in multi-access systems with maximum delay
guarantees
Ismail Cem Atalay.
Proposal:
NUM problem with proportional fairness subject to stability and QoS constraints (delay)
Base station – Access Point
link1 with
channel Ht,1
link2 with
channel Ht,2
linkN with
channel Ht,N
. . . . . . .
End-user terminal 1
End-user terminal 2
End-user terminal N
Lmax(i,1)
Lmax(i,2)
Lmax(i,N)
for inelastic flow i
using link 1
for inelastic flow i
using link 2
for inelastic flow i
using link N
Proposal:
NUM problem with proportional fairness subject to stability and QoS constraints (delay)

Find optimal transmission and resource admission schedule that will
maximixe aggregate end-user utility by also meeting ingress queue stability
criteria for elastic traffic and delay based QoS criteria for inelastic traffic
Maximize aggregate utility for all flows in flow set
(Cel  Cin) forwarded over link set L={l1,l2,...,lN} with
ri as long-term average data rate of flow i.
Subject to below constraints :
•
QOS CONSTRAINT : Maximum allowable ingress backlog size criteria (Lmax) for inelastic traffic.
•
PROPORTIONAL FAIRNESS CONSTRAINT : Utility functions for network flows will be chosen
accordingly so that proportional fairness between end-user terminal flows is satisfied.
•
At each time instant t , a transmission decision will be given on which end-user node will transmit .
The channel capacity of the end-user to BS link will define how many packets will be transmitted..
At each time instant t, a flow-control decision will be given on how much exogenous traffic will be
admitted to end-user terminals.
Physical and virtual backlog sizes are managed by the Lyapunov optimization framework and
should satisfy maximum time delay criteria (Lmax) for inelastic traffic and physical queue stability
criteria for elastic traffic.
•
•
Proposal:
NUM problem with proportional fairness subject to stability and QoS constraints (delay)





Only maximum delay requirements as QoS for inelastic traffic. The criteria for
elastic traffic is stable ingress backlog..
Time-varying wireless channel link capacity is included to scheduling decision
Concave, increasing utility function (logarithmic) for elastic and inelastic traffic
flows providing proportional fairness.
Admission and scheduling decisions are not dependent on any packet arrival
rate/patterns.
Input reservoirs for inelastic flows should be emptied if arrival rates are within
the capacity region of the system.
Traffic Utility function
Proposal:
Example multi-access network with N links and 2N flows



Each end-user terminal is
assumed to produce two types of
traffic flow namely elastic and
inelastic traffic flows.
Each flow i has separate input
reservoir and ingress buffer in
end-user terminals.
Inelastic flows should fulfill an
additional criteria of maximum
allowable time delay. This criteria
can be represented as an upperbound on inelastic flow ingress
queue size.
Optimization framework:






Xi represents transmissions for flow i and Ri stands for exogenous arrivals to
ingress buffer of flow i.
Lmax : maximum backlog size aimed
Ui : The actual physical backlog size for traffic flow buffers.
Cel and Cin : set of elastic and inelastic traffic flows respectively.
ri : the long-term average data rate for flow i
Gi(ri) : the network utility obtained by flow i having data rate ri.
,
Proposed methodology:

Maximum allowable backlog size constraint for inelastic traffic flows could also
be written in below form by assuming E[Xi(t)]=E[Ri(t)]

This corresponds to below virtual queue dynamics :

Physical queue dynamics for inelastic and elastic traffic flows are provided
below :
Lyapunov function and drift :

We aim to have stability in physical and virtual queues. Virtual queue stability
will also satisfy maximum allowed delay constraint for inelastic traffic flows
Below Lyapunov function is proposed to achieve this objective

Lyapunov drift for above Lyapunov function can be written as :

Bounding Lyapunov drift

We have the upper bound for Ui related Lyapunov drift as per below

With constant part upper bounded by

Upper bound Zi related Lyapunov drift is as per below

With constant part upper bounded by
Bounding Lyapunov drift
Utility function
incorporated to Lyapunov drift

To ensure proportional fairness, standard logarithmic function has been utilized
for network utility that depends on instantaneous packet admission rate.

The utility function is incorporated into Lyapunov drift equation via a parameter V
that manages the trade-off between optimal network utility and queue sizes
Lyapunov Drift
RHS maximization

In order to attain stability, Lyapunov drift upper-bound expression on the RHS of
Lyapunov drift inequality should be minimized.

Since we aim to achieve our optimization goals by controlling Xi(t) and Ri(t)
values for flow i, for each step t, admission and transmission control policies that
will minimize Lyapunov drift upper bound should be followed.

Xi (t) represents transmissions for flow i and Ri(t) stands for exogenous arrivals
to ingress buffer of flow i.
Proposed algorithm:
Transmission decision based on minimizing Lyapunov drift
Transmission control decision the coefficient of Xi
So transmission rule is transmit flow itr such that

where Fi(t) is instantaneous channel capacity for the wireless link utilized by flow
i at timeslot t. For the elastic traffic flows Zi(t) is accepted as zero.
Transmission control decision the coefficient of Xi
Proposed algorithm:
Admission decision based on minimizing Lyapunov drift
Admission control decision the coefficient of Ri
So admission rule is admit Ri (t) for inelastic flows such that
And admission rule is admit Ri (t) for elastic flows such that
Transmission control decision the coefficient of Xi
Stability of physical and virtual queues:
Intuitive proof





For Ui(t) < Lmax case, (for any inelastic traffic flow i) Lmax- Ui(t) > 0, so virtual
queue size Zi will always be pulled towards zero according to virtual queue
dynamics
For Ui(t) > Lmax case, (for any inelastic traffic flow i) Zi(t) tends to increase
according to virtual queue dynamics.
However, according to admission rule for inelastic traffic flows, increasing Zi(t)
will curb the admission levels as below condition becomes harder to satisfy
According to scheduling policy (MW based) , the inelastic flows having higher
Zi(t) values will be favored in transmission decisions thus their Xi will increase
So the net effect of increase in Zi(t) is decreasing Ri(t) and increasing Xi(t) for
the respective inelastic flow i.
Transmission control decision the coefficient of Xi
Stability of physical and virtual queues:
Intuitive proof

According to physical queue dynamics equation below
decreasing Ri(t) and increasing Xi(t) will have a decreasing effect on Ui(t).
 Therefore Ui(t) for inelastic traffic flow i will move towards Ui(t) < Lmax region
enforcing the desired upper-bound on physical queue size of inelastic flow.
 Decreasing Ri(t) and increasing Xi(t) will also have a decreasing effect on virtual
queue size Zi(t) according to virtual queue dynamics shown below

Moving towards (Ui(t) < Lmax) region also contributes to the stabilization Zi(t) as
(Lmax - Ui(t)) becomes positive in this case. Therefore virtual queue size stability
in Ui(t) > Lmax case is also satisfied.