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\begin{document}
\title{Markov Chains}
\author{Norman Matloff }
\date{October 31, 2004 \\
\copyright{} 2000-2004, Norman S. Matloff}
\maketitle
\tableofcontents{}
\newpage
\section{Discrete-Time Markov Chains}
\subsection{Introductory Example}
One of the most commonly used stochastic models is that of a
\textbf{Markov chain}. To motivate this discussion, let us start with a
simple example: Consider a \textbf{random walk} on the set of integers
between 1 and 5, moving randomly through that set, say one move per
second, according to the following scheme. If we are currently at
position i, then one time period later we will be at either i-1, i or
i+1, according to the outcome of rolling a fair die---we move to i-1 if
the die comes up 1 or 2, stay at i if the die comes up 3 or 4, and move
to i+1 in the case of a 5 or 6. For the special cases of i = 1 and i =
5, we simply move to 2 or 4, respectively.
The integers 1 through 5 form the \textbf{state space} for this process;
if we are currently at 4, for instance, we say we are in state 4.
Letting $X_{t}$ represent the position of the particle at time t, t =
0, 1,2,\ldots{}which is called the \textbf{state} of the process at time
t.
The random walk is a \textbf{Markov process}. The term \textit{Markov} here
means ``memoryless,'' meaning that we can ``forget the past'':
\begin{equation}
P(X_{t+1}=s_{t+1}|X_{t}=s_{t},X_{t-1}=s_{t-1},\ldots ,X_{0}=s_{0})=P(X_{t+1}=s_{t+1}|X_{t}=s_{t})
\end{equation}
Although this equation has a very complex look, it has a very simple meaning:
The distribution of our next position, given our current position and all our
past positions, is dependent only on the current position. It is clear that
the random walk process above does have this property; for instance, if we are
now at position 4, the probability that our next state will be 3 is 1/3---no
matter where we were in the past.
Continuing this example, let $p_{ij}$ denote the probability of going
from position i to position j in one step. For example,
$p_{21}=p_{23}=\frac{1}{3}$ while $p_{24}=0$ (we can reach position 4
from position 2 in two steps, but not in one step). The numbers $p_{ij}$
are called the \textbf{one-step transition probabilities} of the
process. Denote by P the matrix whose entries are the $p_{ij}$. So, for
instance, row 4 of P in our random walk example is (0,0,1/3,1/3,1/3).
By the way, it turns out that the matrix $P^{k}$ gives the k-step
transition probabilities. In other words, the element (i,j) of this
matrix gives the probability of going from i to j in k steps.
\subsection{Long-Run Distribution}
In typical applications we are interested in the long-run distribution of the
process, for example the long-run proportion of the time that we are at position
4. For each state i, define
\begin{equation}
\label{limnt}
\pi_{i}=\lim_{t\rightarrow \infty }\frac{N_{it}}{t}
\end{equation}
where $N_{it}$ is the number of visits the process makes to state i
among times 1, 2,..., t. In most practical cases, this proportion will
exist and be independent of our initial position $X_{0}$. The $\pi_i$
are called the {\bf steady-state probabilities}, or the {\bf stationary
distribution} of the Markov chain.
Intuitively, the existence of $\pi_{i}$ implies that as t approaches
infinity, the system approaches steady-state, in the sense that
\begin{equation}
\label{limxt}
\lim_{t\rightarrow \infty }P(X_{t}=i)=\pi_{i}
\end{equation}
Actually, the limit (\ref{limxt}) may not exist in some cases. We'll
return to that point later, but for ``typical'' cases it does exist, and
the point is that it suggests a way to calculate the values $\pi_{i}$,
as follows.
First note that
\begin{equation}
P(X_{t+1}=i)
= \sum_{k} P(X_{t}=k \textrm{ and } X_{t+1}=i)
= \sum_{k} P(X_{t}=k) P(X_{t+1}=i | X_t = k)
= \sum_{k} P(X_{t}=k)p_{ki}
\end{equation}
Then as $t\rightarrow \infty $ in this equation, intuitively we
would have
\begin{equation}
\pi_{i}=\sum_{k}\pi_{k}p_{ki}
\end{equation}
Solving these equations (one for each i), called the {\bf balance
equations}, give us the $\pi_i$.
A matrix formulation is also useful. Letting $\pi $ denote the row
vector of the elements $\pi_{i}$, i.e. $\pi = (\pi_1, \pi_2, ...)$,
these equations (one for each i) then have the matrix form
\begin{equation}
\label{pieqn}
\pi =\pi P
\end{equation}
Note that there is also the constraint
\begin{equation}
\sum_{i}\pi_{i}=1
\end{equation}
This can be used to calculate the $\pi_{i}$. For the random walk
problem above, for instance, the solution is $\pi
=(\frac{1}{11},\frac{3}{11},\frac{3}{11},\frac{3}{11},\frac{1}{11})$.
Thus in the long run we will spend 1/11 of our time at position 1, 3/11
of our time at position 2, and so on.
Note that Equation (\ref{pieqn}) may not be easy to solve. For instance,
if the state space is infinite, then this matrix equation represents
infinitely many scalar equations. In such cases, you may need to try to
find some clever trick which will allow you to solve the system, or in
many cases a clever trick to analyze the process in some way other than
explicit solution of the system of equations.
And even for finite state spaces, the matrix may be extremely large. In
some cases, you may need to resort to numerical methods, or symbolic
math packages.
Note again that even if Equation (\ref{pieqn}) has a solution, this does
not imply that (\ref{limxt}) holds. For instance, suppose we alter the
random walk example above so that
\begin{equation}
p_{i,i-1}=p_{i,i+1}=\frac{1}{2}
\end{equation}
for i = 2, 3, 4. In this case, the solution to Equation (\ref{pieqn}) is
$(\frac{1}{8},\frac{1}{4},\frac{1}{4},\frac{1}{4},\frac{1}{8})$. This
solution is still valid, in the sense that Equation (\ref{limnt}) will
hold. For example, we will spend 1/4 of our time at Position 4 in the
long run. But the limit of $P(X_{i}=4)$ will not be 1/4, and in fact
the limit will not even exist. If say $X_{0}$ is even, then $X_{i}$ can
be even only for even values of i. We say that this Markov chain is
{\bf periodic} with period 2, meaning that returns to a given state can
only occur after amounts of time which are multiples of 2.
\subsection{A More Advanced Example}
\subsubsection{Description}
In the above example, the labels for the states consisted of single
integers i. In some other examples, convenient labels may be r-tuples,
for example 2-tuples (i,j).
Consider a serial communication line. Let $B_{1},B_{2},B_{3},...$ denote
the sequence of bits transmitted on this line. It is reasonable to assume the
$B_{i}$ to be independent, and that $P(B_{i}=0)$ and $P(B_{i}=1)$ are
both equal to 0.5.
Suppose that the receiver will eventually fail, with the type of failure
being \textbf{stuck at 0}, meaning that after failure it will report all
future received bits to be 0, regardless of their true value. Once
failed, the receiver stays failed, and should be replaced. Eventually
the new receiver will also fail, and we will replace it; we continue
this process indefinitely.
Let $\rho$ denote the probability that the receiver fails on any given
bit, with independence between bits in terms of receiver failure. Then
the lifetime of the receiver, that is, the time to failure, is
geometrically distributed with ``success'' probability $\rho $ i.e. the
probability of failing on receipt of the i-th bit after the receiver is
installed is $(1-\rho)^{i-1}\rho$ for i = l,2,3,...
However, the problem is that we will not know whether a receiver has
failed (unless we test it once in a while, which we are not including in
this example). If the receiver reports a long string of 0s, we should
suspect that the receiver has failed, but of course we cannot be sure
that it has; it is still possible that the message being transmitted
just happened to contain a long string of 0s.
Suppose we adopt the policy that, if we receive k consecutive 0s, we
will replace the receiver with a new unit. Here k is a design parameter;
what value should we choose for it? If we use a very small value, then
we will incur great expense, due to the fact that we will be replacing
receiver units at an unnecessarily high rate. On the other hand, if we
make k too large, then we will often wait too long to replace the
receiver, and the resulting error rate in received bits will be sizable.
Resolution of this tradeoff between expense and accuracy depends on the
relative importance of the two. (There are also other possibilities,
involving the addition of redundant bits for error detection, such as
parity bits. For simplicity, we will not consider such refinements here.
However, the analysis of more complex systems would be similar to the
one below.)
\subsubsection{Initial Analysis}
A natural state space in this example would be
\begin{equation}
\{(i,j):i=0,1,...,k-1;j=0,1;i+j\neq 0\}
\end{equation}
where i represents the number of consecutive 0s that we have received so
far, and j represents the state of the receiver (0 for failed, 1 for
nonfailed). Note that when we are in a state of the form (k-1,j), if we
receive a 0 on the next bit (whether it is a true 0 or the receiver has
suddenly failed), our new state will be (0,1).
The calculation of the transition matrix P is straightforward, though it
requires careful thought. For example, suppose the current state is
(2,1), and that we are investigating the expense and bit accuracy
corresponding to a policy having k = 5. What can happen upon receipt of
the next bit? The next bit will have a true value of either 0 or 1, with
probability 0.5 each. The receiver will change from working to failed
status with probability $\rho$. Thus our next state could be:
\begin{itemize}
\item (3,1), if a 0 arrives, and the receiver does not fail;
\item (0,1), if a 1 arrives, and the receiver does not fail; or
\item (3,0), if the receiver fails
\end{itemize}
The probabilities of these three transitions out of state (2,1) are:
\begin{eqnarray}
p_{(2,1),(3,1)} & = & 0.5(1-\rho ) \\
p_{(2,1),(0,1)} & = & 0.5(1-\rho ) \\
p_{(2,1),(3,0)} & = & \rho
\end{eqnarray}
Other entries of the matrix P can be computed similarly.
Formally specifying the matrix P using the 2-tuple notation would be
very cumbersome. In this case, it would be much easier to map to a
one-dimensional labeling. For example, if k = 5, the nine states
(1,0),...,(4,0),(0,1),(1,1),...,(4,1) could be renamed states 1,2,...,9.
Then we could form P under this labeling, and the transition
probabilities above would appear as
\begin{eqnarray}
p_{78} & = & 0.5(1-\rho )\\
p_{75} & = & 0.5(1-\rho )\\
p_{73} & = & \rho
\end{eqnarray}
\subsubsection{Going Beyond Finding
\protect\protect\protect\protect\protect$\pi \protect \protect \protect
\protect \protect$}
Finding the $\pi_{i}$ should be just the first step. We then want to
use them to calculate various quantities of interest.\footnote{Note that
unlike a classroom setting, where those quantities would be listed for
the students to calculate, in research we must decide on our own which
quantities are of interest.} For instance, in this example, it would
also be useful to find the error rate $\epsilon$, and the mean time
(i.e., the mean number of bit receptions) between receiver replacements,
$\mu$. We can find both $\epsilon$ and $\mu$ in terms of the $\pi_{i}$,
in the following manner.
The quantity $\epsilon$ is the proportion of the time during which the
true value of the received bit is 1 but the receiver is down, which is
0.5 times the proportion of the time spent in states of the form (i,0):
\begin{equation}
\epsilon =0.5(\pi_{1}+\pi_{2}+\pi_{3}+\pi _{4})
\end{equation}
Now to get $\mu$ in terms of the $\pi_{i}$ note that since $\mu$ is the
mean number of bits between receiver replacements, it is then the
reciprocal of the proportion of bits that result in replacements. For
example, if 5\% of the received bits result in replacement of the
receiver, then (speaking on an intuitive level) the ``average'' set of
20 bits will contain one bit which makes us replace the receiver, and
there will thus be an average of 20 bits between replacements.
Again suppose k = 5. A replacement will occur only from states of the
form (4,j), and even then only under the condition that the next
reported bit is a 0. In other words, there are three possible ways in
which replacement can occur:
\begin{itemize}
\item [(a)] We are in state (4,0). Here, since the receiver has failed,
the next reported bit will definitely be a 0, regardless of that bit's
true value. We will then have a total of k = 5 consecutive received 0s,
and therefore will replace the receiver.
\item [(b)] We are in the state (4,1), and the next bit to arrive is a
true 0. It then will be reported as a 0, our fifth consecutive 0, and we
will replace the receiver, as in (a).
\item [(c)] We are in the state (4,1), and the next bit to arrive is a
true 1, but the receiver fails at that time, resulting in the reported
value being a 0. Again we have five consecutive reported 0s, so we
replace the receiver.
\end{itemize}
Therefore,
\begin{equation}
\mu ^{-1}=\pi_{4}+\pi_{9}(0.5+0.5\rho )
\end{equation}
and thus
\begin{equation}
\mu =\frac{1}{\pi_{4}+0.5\pi_{9}(1+\rho )}
\end{equation}
This kind of analysis could be used as the core of a cost-benefit
tradeoff investigation to determine a good value of k. (Note that the
$\pi_{i}$ are functions of k, and that the above equations for the case
k = 5 must be modified for other values of k.)
\subsection{A Shared-Memory Multiprocessor Example}
(Adapted from \textit{Probabiility and Statistics, with Reliability, Queuing
and Computer Science Applicatiions}, by K.S. Trivedi, Prentice-Hall, 1982.)
\subsubsection{The Model}
Consider a shared-memory multiprocessor system with m memory modules and
m CPUs. The address space is partitioned into m chunks, based on either
the most-significant or least-significant $\log_2m$ bits in the
address.\footnote{Many of you may recognize this as high-order and
low-order interleaving, respectively.}
The CPUs will need to access the memory modules in some random way,
depending on the programs they are running. To make this idea concrete,
consider the Intel assembly language instruction
\begin{verbatim}
add %eax, (%ebx)
\end{verbatim}
which adds the contents of the EAX register to the word in memory
pointed to by the EBX register. Execution of that instruction will
(absent cache and other similar effects, as we will assume here and
below) involve two accesses to memory---one to fetch the old value of
the word pointed to by EBX, and another to store the new value.
Moreover, the instruction itself must be fetched from memory. So,
altogether the processing of this instruction involves three memory
accesses.
Since different programs are made up of different instructions, use
different register values and so on, the sequence of addresses in memory
that are generated by CPUs are modeled as random variables. In our
model here, the CPUs are assumed to act independently of each other, and
successive requests from a given CPU are independent of each other too.
A CPU will choose the i-th module with probability $q_{i}$. A memory
request takes one unit of time to process, though the wait may be longer
due to queuing. As soon as a CPU's memory request is fulfilled, it
generates another one.
How good are these assumptions? One weakness, for instance, is that
many instructions, for example, do not use memory at all, except for the
instruction fetch, and as mentioned, even the latter may be suppressed
due to cache effects.
As another example, many programs will have code like
\begin{verbatim}
for (i = 0; i < 10000; i++) sum += x[i];
\end{verbatim}
Since the elements of the array x will be stored in consecutive
addresses, successive memory requests from the CPU while executing this
code will not be independent.
Thus, many models of systems like this have been quite complex, in order
to capture the effects of various things like caching, nonindependence
and so on in the model. Nevertheless, one can often get some insight
from even very simple models too. In any case, for our purposes here it
is best to stick to simple models, so as to understand more easily.
Our state will be an m-tuple $(n_{1},...,n_{m})$, where $n_{i}$
is the number of requests currently pending at memory module i. Assume
that a CPU generates another memory request almost immediately after the
previous one is fulfilled. This implies that $n_{1}+...+n_{m}=m$.
It is straightforward to find the transition probabilities $p_{ij}$.
For example, suppose m = 2 and we want to find~\texttt{$
p_{(2,0),(1,1)}$.} For the transition $(2,0)\rightarrow (1,1)$ to occur,
when the request being served at Module 1 is done (currently there are 2
requests pending at that module, the one being served and the one in the
queue), the CPU which requested it will submit a new request, this time
for Module 2. This will occur with probability $q_{2}$. (Meanwhile, the
request which had been queued at Module 1 will now start service.) So,
$p_{(2,0),(1,1)}=q_{2}$.
\subsubsection{Going Beyond Finding the \protect\protect\protect$\pi_{i}\protect \protect \protect$}
Let B denote the number of memory requests completed per unit time. We
can find E(B) as follows. Let S denote the current state. Then,
continuing the case m = 2, we have
\begin{eqnarray}
E(B) & = & E[E(B|S)] \nonumber \\
& = & \pi_{(2,0)}E(B|S=(2,0))+\pi_{(1,1)}E(B|S=(1,1))+\pi_{(0,2)}E(B|S=(0,2)) \nonumber \\
& = & ... \nonumber \\
& = & \frac{1-q_{1}q_{2}}{1-2q_{1}q_{2}}
\end{eqnarray}
For example E(B$|$S)=(2,0) = 1. The other computations above are
similar.
The maximum value of E(B) occurs when $q_{1}=q_{2}=\frac{1}{2}$,
which case E(B)=1.5. This is a lot less than the maximum capacity of the
memory system, which is m = 2 requests per cycle.
\subsection{A Slotted ALOHA Example}
(Famous example, adapted from various sources.)
\subsubsection{Resolving Problems with Finding the
\protect\protect\protect$\pi_{i}\protect \protect \protect $}
As pointed out earlier, solving the set of Equations (\ref{pieqn}) may
be a problem. This is likely in the case that the state space is
infinite, but even is true for finite state spaces. A state space can
easily contain thousands of states, which would mean many megabytes of
storage space (thus at least affecting solution time, due to virtual
memory page faults), and likely means serious roundoff error problems.
However, in some cases there is some kind of structure to the matrix P
which can be exploited to facilitate the solution of the equations. In
this subsection we will develop an example of this.
Consider a simple form of a \textbf{slotted ALOHA} communications
channel. There are n nodes, each of which is either idle or has a single
message transmission pending. Time is \textbf{slotted}, i.e. a node is
allowed to transmit only at integer-valued time epochs. Just slightly
after the beginning of each time slot, each of the idle nodes generates
a message with probability p. If more than one node attempts to send
within a given time slot, there is a \textbf{collision}, and each of the
transmissions involved will fail. So, we include a {\bf backoff}
mechanism: At the middle of each time slot, each node with a message
will with probability q attempt to send the message, with the
transmission time occupying the remainder of the slot. While a node has
a message transmission pending, it does not generate any new messages.
So, q is a design parameter, which must be chosen carefully. If q is
too large, we will have too mnay collisions, thus increasing the average
time to send a message. If q is too small, a node will often refrain
from sending even if no other node is there to collide with.
Define our state for any given time slot to be the number of nodes currently
having a message to send at the very beginning of the time slot (before new
messages are generated). Then for $0 < i < n$ and $j > 0$, we have
\begin{eqnarray}
p_{i,i-1} & = & (1-p)^{n-i}i(1-q)^{i-1}q\label{aloha} \\
p_{ii} & = & (1-p)^{n-i}[1-i(1-q)^{i-1}q]+(n-i)(1-p)^{n-i-1}p(i+1)(1-q)^{i}q\\
p_{i,i+j} & = & C(n-i,j)p^{j}(1-p)^{n-i-j}[1-(i+j)(1-q)^{i+j-1}q]\\
& + & C(n-i,j+1)p^{j+1}(1-p)^{n-i-j-1}(i+j+1)(1-q)^{i+j}q
\end{eqnarray}
where C(r,s) is the number of combinations of r things taken s at a
time:
\begin{equation}
C(r,s) = \binom{r}{s}
\end{equation}
For example, the equation for $p_{i,i+j}$ reflects the fact that there
are two ways to get to state i+j from state i: j of the n-i idle states could
generate messages and then none of the i+j nodes with messages successfully
transmits its message, or j+1 of the idle nodes generate messages and then one
of the i+j+1 nonidle nodes successfully generates a message.
The matrix P is then rather complex. Note, however, that the entries
$p_{i,i-j}$ for $j > 1$ are all 0s. This suggests that a way to ease our
computational burden. Set
\begin{eqnarray}
r_{i} & = & p_{i,i-1}\\
s_{i} & = & p_{ii}\\
t_{ij} & = & p_{i,i+j}
\end{eqnarray}
for $0r+s|W>r)=P(W>s)
\end{equation}
for all positive r and s. Actually, one can show that exponential
distributions are the only continuous distributions which have this
property. Therefore, \textsl{holding times in Markov chains must be
exponentially distributed.}
It is difficult for the beginning modeler to fully appreciate the
memoryless property. Consider the problem of waiting to cross the
railroad tracks on Eighth Street in Davis, just west of J Street. One
cannot see down the tracks, so we don't know whether the end of the
train will come soon or not.
If we are driving, the issue at hand is whether to turn off the car's
engine. If we leave it on, and the end of the train does not come for a
long time, we will be wasting gasoline; if we turn it off, and the end
does come soon, we will have to start the engine again, which also
wastes gasoline.
Suppose our policy is to turn off the engine if the end of the train
won't come for at least s seconds. Suppose also that we arrived at the
railroad crossing just when the train first arrived, and we have already
waited for r seconds. Will the end of the train come within s more
seconds? If the length of the train were exponentially distributed,
Equation (\ref{nomem}) would say that the fact that we have waited r
seconds so far is of no value at all in predicting whether the train
will end within the next s seconds. The chance of it lasting at least s
more seconds right now is no more and no less than the chance it had of
lasting at least s seconds when it first arrived.
Because it is key to the Markov property, the exponential distribution
is assumed for all basic activities in Markov models. In queuing models,
for instance, both the interarrival time and service time are assumed to
be exponentially distributed (though of course with different values of
$\lambda$). In reliability modeling, the lifetime of a component is
assumed to have an exponential distribution.
Such assumptions have in many cases been verified empirically. If you go to
a bank, for example, and record data on when customers arrive at the
door, you will find the exponential model to work well (though you may
have to restrict yourself to a given time of day, to account for
nonrandom effects such as heavy traffic at the noon hour). In a study of
time to failure for airplane air conditioners, the distribution was also
found to be well fitted by an exponential density. On the other hand,
in many cases the distribution is not close to exponential, and purely
Markovian models cannot be used.
\subsection{Analysis}
We again define $\pi_{i}$ to be the long-run proportion of time the
system is in State i, and we again will derive a system of linear
equations to solve for these proportions.
To this end, let $\lambda_{i}$ denote the parameter in the
holding-time distribution at State i, and define $\rho_{ji}$ to be
the number of transitions from State j to State i per unit time, during
times that we are in State j.\footnote{In other words,
$$
\rho_{ji} = \lim_{{t} \rightarrow \infty} \frac{C_{ji}(t)}{D_j(t)}
$$
\noindent where $C_{ji}(t)$ is the number of j-to-i transitions during
(0,t) and $D_j(t)$ is the amount of time the process spends at j during
(0,t).} In the long run, the number of transitions into State i per unit
time must equal the number of transitions out of it. Let us calculate
(and then equate) these two quantities:
For a given state j, the long-run number of transitions into State i
from State j per unit time is $\pi_{j}\rho_{ji}$. (We are in State
j a proportion $\pi_{j}$ of the time, and during that time
transitions are made into State i at a rate of $\rho_{ji}$.) So, the
total number of transitions into i is
\begin{equation}
\sum_{j\neq i}\pi_{j}\rho_{ji}
\end{equation}
What about the number of transitions out of i? Since the mean holding
time is $1/\lambda_{i}$, that means that during the times we are in
State i, transitions out will occur at rate $\lambda_{i}$. Thus
\begin{equation}
\label{contin1}
\sum_{j\neq i}\pi_{j}\rho_{ji}=\pi_{i}\lambda_{i}
\end{equation}
Note further that if we let $p_{ji}$ denote the probability that, when a
transition out of j occurs, it is to State i, then $\rho_{ji}=\lambda
_{j}p_{ji}$. So, an alternate form of Equation (\ref{contin1}) is
\begin{equation}
\sum_{j\neq i}\pi_{j}\lambda_{j}p_{ji}=\pi_{i}\lambda_{i}
\end{equation}
\subsection{\label{min}Minima of Independent Exponentially Distributed
Random Variables}
Suppose $W_{1},...,W_{k}$ are independent random variables, with $W_{i}$
being exponentially distributed with parameter $\lambda_{i}$. Let
$Z=min(W_{1},...,W_{k})$. Then
\begin{itemize}
\item Z is exponentially distributed with parameter $\lambda
_{1}+...+\lambda_{k}$
\item $P(Z=W_{i}) = \frac{\lambda_{i}}{\lambda_{1}+...+\lambda_{k}}$
\end{itemize}
These facts are easy to prove, starting with the relation
\begin{equation}
F_{Z}(t)=1-P(Z>t)=1-P(W_{1}>t\textrm{ and }...\textrm{ and }W_{k}>t)=
1-\Pi_{i}\textrm{ }e^{-\lambda_{i}t}
\end{equation}
The details are left to the reader.
\subsection{Machine Repair Example}
Suppose the operations in a factory require the use of a certain kind of
machine. The manager has installed two of these machines. This is known
as a \textbf{gracefully degrading system}: When both machines are
working, the fact that there are two of them, instead of one, leads to a
shorter wait time for access to a machine. When one machine has failed,
the wait is longer, but at least the factory operations may continue. Of
course, if both machines fail, the factory must shut down until at least
one machine is repaired.
Suppose the time until failure of a single machine, carrying the full load of
the factory, has an exponential distribution with mean 20.0, but the mean is
25.0 when the other machine is working, since it is not so loaded. Repair time
is exponentially distributed with mean 8.0.
We can take as our state space \{0,1,2\}, where the state is the number
of working machines. Now, let us find the parameters $\lambda_{i}$ and
$\rho_{ji}$ for this system. For example, what about $\lambda_{2}$?
The holding time in State 2 is the minimum of the two lifetimes of the
machines, and thus from the results of Section \ref{min}, has parameter
$\frac{1}{25.0}+\frac{1}{25.0}=0.08$.
For $\lambda_{1}$, a transition out of State 1 will be either to State
2 (the down machine is repaired) or to State 0 (the up machine fails).
The time until transition will be the minimum of the lifetime of the up
machine and the repair time of the down machine, and thus will have
parameter $\frac{1}{20.0}+\frac{1}{8.0}=0.175$. Similarly, $\lambda
_{0}=\frac{1}{8.0}+\frac{1}{8.0}=0.25$.
It is important to understand how the Markov property is being used
here. Suppose we are in State 1, and the down machine is repaired,
sending us into State 2. Remember, the machine which had already been
up has ``lived'' for some time now. But the memoryless property of the
exponential distribution implies that this machine is now ``born
again.''
What about the parameters $\rho_{ji}$? Well, $\rho_{21}$ is certainly
easy to find; since the transition $2\rightarrow 1$ is the \textit{only}
transition possible out of State 2, then $\rho_{21}$ is the rate of
\textit{all} transitions out of State 2, i.e. $\lambda_{2}=0.08$.
What about $\rho_{12}$? This is the $\frac{1}{8.0}$ portion of the
transition rate out of State 1, $\frac{1}{20.0}+\frac{1}{8.0}$, i.e. 0.125.
Working in this manner, we finally arrive at the complete system of equations
(\ref{contin1}):
\begin{eqnarray}
\pi_{2}(0.08) & = & \pi_{1}(0.125)\\
\pi_{1}(0.175) & = & \pi_{2}(0.08)+\pi_{0}(0.25)\\
\pi_{0}(0.25) & = & \pi_{1}(0.05)
\end{eqnarray}
Of course, we also have the constraint $\pi_{2}+\pi_{1}+\pi_{0}=1$.
The solution turns out to be
\begin{equation}
\pi =(0.072,0.362,0.566)
\end{equation}
Thus for example, during 7.2\% of the time, there will be no machine available
at all.
Several variations of this problem could be analyzed. We could compare
the two-machine system with a one-machine version. It turns out that the
proportion of down time (i.e. time when no machine is available)
increases to 28.6\%. Or we could analyze the case in which only one
repair person is employed by this factory, so that only one machine can
be repaired at a time, compared to the situation above, in which we
(tacitly) assumed that if both machines are down, they can be repaired
in parallel. We leave these variations as exercises for the reader.
\subsection{Continuous-Time Birth/Death Processes}
We noted earlier that the system of equations for the $\pi_i$ may not be
easy to solve. In many cases, for instance, the state space is infinite
and thus the system of equations is infinite too. However, there are
some rich classes of Markov chains for which closed-form solutions have
been found.
One such class is \textbf{birth/death processes}.\footnote{Though we
treat the continuous-time case here, there is also a discrete-time
analog.} Here the state space consists of (or has been mapped to) the
set of nonnegative integers, and $\rho_{ji}$ is nonzero only in cases in
which $|i-j| = 1$. (The name ``birth/death'' has its origin in Markov
models of biological populations, in which the state is the current
population size.) Note for instance that the example of the gracefully
degrading system above has this form. An M/M/1 queue---one server,
``Markov'' (i.e. exponential) interarrival times and Markov service
times---is also a birth/death process, with the state being the number
of jobs in the system.
Here is the solution. Let
\begin{equation}
r_{i}=\Pi ^{i}_{k=1}\frac{\rho_{k-1,k}}{\rho_{k,k-1}}
\end{equation}
where $r_i = 0$ for $i > m$ if the chain has no states past m.
Then
\begin{equation}
\pi_{n}=\pi_{0}r_{n}\textrm{ }(n>0)
\end{equation}
where
\begin{equation}
\pi_{0}=\frac{1}{1+\sum ^{\infty }_{i=1}r_{i}}
\end{equation}
\section{Some Mathematical Conditions}
There is a rich mathematical theory regarding the asymptotic behavior of
Markov chains. We will not present such material here in this brief
introduction, but we will give an example of the implications the theory
can have.
\subsection{Some Definitions and Theorems}
A state in a Markov chain is called \textbf{recurrent} if it is
guaranteed that, if we start at that state, we will return to the state
infinitely many times. A nonrecurrent state is called
\textbf{transient.}
Let $T_{ii}$ denote the time needed to return to state i if we start
there.\footnote{Keep in mind that $T_{ii}$ is the time from one entry to
state i to the next entry to state i. So, it includes time spent in
i---1 for a discrete-time chain, a random exponential amount of time in
the continuous-time case---then time spent away from i, up to the time
of next entry to i.} Note that an equivalent definition of recurrence is
that $P(T_{ii}<\infty )=1$, i.e. we are sure to return to i at least
once. By the Markov property, if we are sure to return once, then we
are sure to return again once after that, and so on, so this implies
infinitely many visits.
A recurrent state i is called \textbf{positive recurrent} if
$E(T_{ii})<\infty$, while a state which is recurrent but not positive
recurrent is called \textbf{null recurrent}. It can be shown that for a
positive recurrent state i in a discrete-time Markov chain,\footnote{An
analogous result holds for continuous-time chains.}
\begin{equation}
\label{etii}
\pi_{i}=\frac{1}{E(T_{ii})}
\end{equation}
We will not prove this here, but please read the intuitive analysis in
Appendix \ref{secondib}.
Thus positive recurrence means that $\pi_{i}>0$. For a null
recurrent chain, the limits in Equation (\ref{limnt}) are 0, which means
that there is very little one can say of interest regarding the long-run
behavior of the chain.
For example, consider the famous \textbf{random walk} on the integers:
At each time step, one goes left one integer or right one integer (e.g.
to +3 or +5 from +4), with probability 1/2 each. It turns out that this
chain (meaning all states in the chain) is recurrent but not null
recurrent. By the way, the same is true for the corresponding random
walk on the two-dimensional integer lattice (moving up, down, left or
right with probability 1/4 each). However, in the three-dimensional
case, the chain is not even null recurrent; it is transient.
Consider an \textbf{irreducible} Markov chain, meaning one which has the
property that one can get from any state to any other state (though not
necessarily in one step). One can show that in an irreducible chain, if
one state is recurrent then they all are.
\subsection{Tree-Searching Example}
(Adapted from {\it Performance Modelling of Communication Networks and
Computer Architectures}, by P. Harrison and N. Patel, pub. by
Addison-Wesley, 1993.)
Consider the following Markov chain with infinite state space
\{0,1,2,3,...\}. The transition matrix is defined by $p_{i,i+1}=q_{i}$
and $p_{i0}=1-q_{i}$. This kind of model has many different
applications, including in computer science tree-searching algorithms.
(The state represents the level in the tree where the search is
currently, and a return to 0 represents a backtrack. More general
backtracking can be modeled similarly.)
The question at hand is, What conditions on the $q_{i}$ will give us
a positive recurrent chain?
Assuming $0n)=\Pi_{i=0}^{n-1}q_{i}
\end{equation}
Therefore, the chain is recurrent if and only if
\begin{equation}
\lim_{n\rightarrow \infty }\Pi_{i=0}^{n-1}q_{i}=0
\end{equation}
For positive recurrence, we need $E(T_{00})<\infty$. Now, one can show
that for any nonnegative integer-valued random variable Y
\begin{equation}
E(Y)=\sum ^{\infty }_{n=0}P(Y>n)
\end{equation}
Thus for positive recurrence, our condition on the $q_i$ is
\begin{equation}
\sum ^{\infty }_{n=0}\Pi ^{n-1}_{i=0}q_{i}<\infty
\end{equation}
\subsection{Computer Worm Example}
A computer science graduate student at UCD, C. Senthilkumar, was working
on a worm alert mechanism. A simplified version of the model is that
network hosts are divided into groups of size g, say on the basis of
sharing the same router. Each infected host tries to infect all the
others in the group. When g-1 group members are infected, an alert is
sent to the outside world.
The student was studying this model via simulation, and found some
surprising behavior. No matter how large he made g, the mean time until
an external alert was raised seemed bounded. He asked me for advice.
I modeled this as a pure birth process. In state i, there are i
infected hosts, each trying to infect all of the g-i noninfected hots.
When the process reaches state g-1, the process ends; we call this state
an {\bf absorbing state}, i.e. one from which the process never leaves.
Suppose that for each infected/noninfected pair of hosts, the time to
infection of the noninfected member by the infected member has an
exponential distribution with mean 1.0. Assume independence among the
various infection attempts. Since in state i there are i(g-i) such
pairs, and since we go to state i+1 when the first infection among these
occurs, we have $\lambda_i = i(g-i)$. Thus the mean time to go from
state i to state i+1 is 1/[i(g-i)].
Then the mean time to go from state 1 to state g-1 is
\begin{equation}
\label{s1}
\sum_{i=1}^{g-1} \frac{1}{i(g-i)}
\end{equation}
Using a calculus approximation, we have
\begin{equation}
\int_1^{g-1} \frac{1}{x(g-x)} ~ dx = \frac{1}{g} \int_1^{g-1}
(\frac{1}{x} +
\frac{1}{g-x}) ~ dx = \frac{2}{g} ln(g-1)
\end{equation}
The latter quantity goes to zero as $g \rightarrow \infty$. This
confirms that the behavior seen by the student in simulations holds in
general. In other words, (\ref{s1}) remains bounded as $g \rightarrow
\infty$. This is a very interesting result, since it says that the mean
time to alert is bounded no matter how big our group size is.
\appendix
\section{Intuitive ``Proof'' of Equation (\ref{etii})}
\label{secondib}
The approach is similar to that of Section \ref{firstib}. Define
alternating On and Off subcycles, where On means we are at State i and
Off means we are elsewhere. An On subcycle has duration 1, and an Off
subcycle has duration $T_{ii}-1$. Define a full cycle to consist of an
On subcycle followed by an Off subcycle.
Then intuitively the proportion of time we are in State i is
\begin{equation}
\pi_i = \frac{E(\textrm{On})}{E(\textrm{On}) + E(\textrm{Off})} = \frac{1}{1+E(T_{ii}-1)} = \frac{1}{ET_{ii}}
\end{equation}
One can form a similar equation for the continuous-time case. Here
$E(\textrm{On}) = 1/\lambda_i$. The Off subcycle has duration
$T_{ii}-1/\lambda_i$. Note that $T_{ii}$ is measured from the time we
enter State i once until the time we enter it again.
\end{document}