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\title{\bf EE 290Q Topics in Communication Networks\\ Lecture Notes: 4}
\author{Soheila Bana}
\date{January 25, 1996}
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{\footnotesize
\subsection*{Preface}
The lecture is in a series on {\em Analysis of the end-to-end
performance of a network }. This lecture focuses on {\em end-to-end service curve of a GPS network}.
Such analysis is helpful in estimating the {\em worst-case end-to-end delay}.
Assuming a leaky bucket regulator at the source, and Consistent Relative
Session Treatment (CRST), the idea of {\em service curve} is used to calculate
a lower bound on the number of packets serviced during a period of time. This calculations assume a stable network.
The material is based on an article by
Parekh and Gallager \cite{Parekh} and an article by Cruz \cite{Cruz}.}
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\section{Introduction}
In a network, it is desired to solve the dynamic problem of user allocation.
However, one needs to know the boundary of the service provided to the users to
approach the problem. The minimum service guaranteed to the user is analyzed
by the use of {\em service curves} which provide a convenient framework for
managing the allocation of performance guarantees. It should be mentioned that
this worst case end to end service curve for GPS assumes perfect isolation
between different nodes, and staggered
greedy regime for all nodes. This in fact is an extreme case ; the importance of it is that it achieves a bound for the service.
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\section{Review Concepts and Notations}
Concepts to be used from last lectures (see Lecture Notes 2 and 3)
\begin{itemize}
\item Leaky Bucket traffic regulator
\item Generalized Processor Sharing (GPS)
\item Service Curve
\item Burstiness constraint
\item Greedy regime
\item ${A^m}_i[s,t]$ :- arrival to node $m$ during interval $[s, t]$
\item ${S^m}_i[s,t]$ :- amount served at node $m$ in $[s,t]$ interval
\item $H_i$ :- number of nodes (hops)
\item $\phi^m_i$ :- service weight given to session $i$ by the network at each node
\item $m$ :- node in the network
\item ${Q^m}_i(t)$ :- backlog of session $i$ at node $m$
\item $Q_i(t)$ :- end to end backlog;
$Q_i(t)$ = $\sum_{m=1}^{H_i} Q_i^m(t)$
\item ${\hat S}_i(\tau)$:-service curve, minimum guaranteed service
\item ${\hat A}_i(\tau)$:-maximum burstiness constraint
\end{itemize}
\section{The End-to-End Service Curve}
By definition, a service curve for a node specifies the minimum guaranteed service when
backlogged in a all-greedy regime in GPS.
An end-to-end service curve is derived from individual service curves. Such a
service curve partially characterizes the service received by a connection at a network.
Notice that this minimum service is not derived from scheduling policies,
but is allocated, and could be used in scheduling policies. Since it is derived
from individual service curves, it assumes complete isolation between different
nodes which is not a realistic assumption. What it means is that in reality we
often have a better service rate than what we see here.
We assume that arrival is regulated by a leaky bucket, and the GPS network is stable,
i.e. every packet will be served in a finite amount of time. Also, there is no priority packet service at a session at this
point and packets are served on a first come first serve basis(FIFO) within a session.
\begin{definition} A network is said to guarantee service curve ${\hat S}_i(\tau)$
for session i if $\forall \ t$, $\exists \ s < t$ such that
$$S_i^{H_i}[0,t] \geq {\hat S}_i[t-s] + A_i[0,s] \eqno(1)$$
\end{definition}
\begin{theorem}
An end-to-end service curve is not unique and can be expressed as:
$${\hat S}_i(t) = min_{\sum_{t_m=t}}\sum_{m=i}^{H_i}{\hat S}_i^m(t_m) \eqno(2)$$
\end{theorem}
\begin{figure}[ht]
\centerline{\ \psfig{figure=fig1.epsi,width=5in}}
\end{figure}
\begin{figure}
\centerline{\ \psfig{figure=fig2.epsi,width=5in}}
\end{figure}
\noindent
{\bf Proof:} We use the definition of the service curve to prove the theorem. Remember that:
$$S_i^{H_i}[0,t] \geq A_i[0,s] + {\hat S}_i[t-s]\eqno(3) $$
Consider that for a fixed time t, you partition the time by $\tau_{H_i-1} < t$
such that
$$S_i^{H_i}[0,t]=S_i^{H_i}[0,\tau_{H_i-1}] + S_i^{H_i}[\tau_{H_i-1},t]\eqno(4)$$
and
$${Q_i^{H_i}}_{(\tau_{{H_i}-1})} = 0\eqno(5)$$
The above condition means that all the packets arrived by time $\tau_{{H_i}-1}$
have been served, so the number of packets arrived for the period of
$[0,\tau_{{H_i}-1}]$ equals the number of packets served.
$$A_i^{H_i}[0,\tau_{H_i-1}]=S_i^{H_i}[0,\tau_{H_i-1}]\eqno(6)$$
and also by definition of nodal service curve,
$$S_i^{H_i}[\tau_{H_i-1},t] \geq {\hat S}_i^{H_i}[t-\tau_{H_i-1}]\eqno(7)$$
By substitution of (6) and (7) in (4) we get a lower bound for $S_i^{H_i}[0,t]$:
$$S_i^{H_i}[0,t] \geq A_i^{H_i}[0,\tau_{H_i-1}] + {\hat S}_i^{H_i}[t-\tau_{H_i-1}]\eqno(8)$$
\begin{figure}[ht]
\centerline{\ \psfig{figure=fig3.epsi,width=5in}}
\end{figure}
Similarly, we can partition the time interval $[0,\tau_{H_i-1}]$ again by $\tau_{H_i-2}$. As a
result, we get a lower bound for $ S_i^{H_i}[0,\tau_{H_i-1}]$:
$$ S_i^{H_i}[0,\tau_{H_i-1}] \geq A_i^{H_i-1}[0,\tau_{H_i-2}] + {\hat S}_i^{H_i-1}[\tau_{H_i-1}-\tau_{H_i-2}]\eqno(9)$$
Again, because at time $\tau_{H_i-2}$ there is no queue and $Q^{Hi_-1}(\tau_{H_i-2}) = 0$. Using (6), and substituting lower bound of $ S_i^{H_i}[0,\tau_{H_i-1}]$
in (9) for $A_i^{H_i}[0,\tau_{H_i-1}]$ in (8) gives
$$S_i^{H_i}[0,t] \geq A_i^{H_i-1}[0,\tau_{H_i-2}] + {\hat S}_i^{H_i-1}[\tau_{H_i-2}-\tau_{H_i-1}] + {\hat S}_i^{H_i}[t- \tau_{H_i-1}]\eqno(10)$$
and continue similarly for $\tau_{H_i-3}$, $\tau_{H_i-4}$, ... to $\tau_0$ and we get:
$$S_i^{H_i}[0,t] \geq A_i^{H_i}[0,\tau_0] + \sum_{m=1}^{H_i}{\hat S}_i^m[\tau_m-\tau_{m-1}] \eqno(11)$$
Now, let $\tau_0 = s$ and compare (3) with (1) and notice that we can conclude
that (2) holds.
\hfill{$\Box$}
\noindent
Once again, it should be mentioned that in the above theorem, we have considered
that all cross traffic is greedy and we get the minimum service and worst case delay.
Such traffic conspiracy by all nodes in not true in general and traffic cannot burst
arbitrarily at all nodes. So the importance of minimum guaranteed service lies in the
achieved boundary.
Also, since for all greedy regime in GPS the service curve ${\hat S}_i^m(t_m)$ for each node is piecewise linear,
the end-to-end service curve ${\hat S}_i(t)$
is a piecewise linear and convex curve.
\section{End-to-End Worst Case Delay}
The less the service the longer the delay is in a network. Therefore, the minimum
guaranteed service not only bounds the minimum service, but also gives a bound
to the longest end-to-end delay. Even though not really achieved, such a delay
depends on
{\em i)} worst-case delay at each node
{\em ii)} staggered fashion all-greedy regime along the path in relation to the
end-to-end service curve.
Assume the minimizing solution to (2) is $\tilde{t_0}$, $\tilde{t_1}$, ... In other words,
at time $0$ first node has the worst traffic by all-greedy regime, while at time $\tilde{t_0}$
the second node has it, and so on. This staggered all greedy regime in GPS achieves
the end-to-end service curve.
Having two curves of the burstiness constraint, ${\hat A}_i$ and the end-to-end service curve,
${\hat S}_i$, the worst case delay is specified by the maximum horizontal distance
between the two curves, as shown in Fig. 4.
\begin{figure}[ht]
\centerline{\ \psfig{figure=fig4.epsi,width=5in}}
\end{figure}
{\em open question:} Given weight assignment for GPS, we can compute performance
measures such as service rate, delay, etc. But given users with delay requirements
, can one compute $\phi^m_i$ to guarantee the requirements?
Such a problem is a difficult one. It is a network design problem to design a scheduling
policy for required delays by users. Usually the problem of resource allocation is
a dynamic one.
\section{Implementation Issue}
GPS is an ideal model of the network, because it assumes the packets are infinitesimally
small and can instantly hop from one node to another. However, packets have finite
length and serving each packet takes some time. That is why Packet by Packet GPS (PGPS) is used to model the networks (see Lecture Note 5).
Though GPS does not consider the serving time of packets, still we can talk about the
order of serving for the packets.
\begin{fact}
Under GPS, the departure order of two packets, say $p$ and $\tilde{p}$,
at time t is independent of future arrivals.
\end{fact}
This is because by definition, GPS serves sessions with regard to their service
weight $\phi^m_i$ and service order is not changed. So, if session i is backlogged,
still its rate of service by the GPS network is independent of the length of
other sessions' queues.
GPS will not be confused about the service order of packets. Confusion may happen
when the packet which should be served has not arrived yet. In this case, GPS
would go ahead and depart the next in line.
\section{Conclusion}
One important issue in analyzing the quality of service in a network is the delay bound
which is inversely proportional to the minimum guaranteed rate of service. The deterministic
quality of service bound was reviewed and calculated in terms of arrival rate and nodal
service curves under the worst conditions. As an interesting subject for future work is
a similar approach in a probabilistic framework. Another idea for future work,
considering the fact that service curves are not unique, is to consider the problem
dynamically, allocate and deallocate the service curve based on the circuit's need.
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\begin{thebibliography}{2}
\bibitem{Cruz}
Cruz, R.L. (1995).
Quality of Service Guarantees in Virtual Circuit Switched Networks. {\em
IEEE Journal on Selected Areas in Communications}, August 1995, Vol. 13,
No. 6, pp. 1048-1056.
\bibitem{Parekh}
Parekh, Abhay K. and Gallager, Robert G. (1993).
A Generalized Processor Sharing Approach to Flow Control in Integrated
Services Networks: the Single-Node Case. {\em IEEE/ACM Transactions on
Networking}, June 1993, Vol. 1, No. 3, pp. 344-357.
\bibitem{Parekh_2}
Parekh, Abhay K. and Gallager, Robert G. (1994).
A Generalized Processor Sharing Approach to Flow Control in Integrated
Services Networks: the Multiple Node Case. {\em IEEE/ACM Transactions on
Networking}, April 1994, Vol. 2, No. 2, pp. 137-150.
\end{thebibliography}
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