We discuss the problem of the optimal
liquidation of a financial product in which both the market risk
of the asset and the market impact of the investor's own dealings
are considered, and where the asset is liquidated over several
sales periods with a constant sales interval. The investor chooses
the sales volume in each period as well as the volume over the
entire sales period in order to minimize the expected execution
costs under a certain level of risk. We obtain an explicit
solution for the optimal execution strategies and present four
numerical examples to show that the proportion between market risk
and liquidity risk exerts a major influence over the optimal
execution strategy. We also show that to obtain the optimal result
the investor should liquidate his holdings over a short sales
In this paper we consider the analysis of a tandem queueing model
$M/G/1 -> ./M/1$. In contrast to the vast majority of the previous
literature on tandem queuing models we consider the case with GI
service time at the first queue and with infinite buffers. The
system can be described by an M/G/1-type Markov process at the
departure epochs of the first queue. The main result of the paper is
the steady-state vector generating function at the embedded epochs,
which characterizes the joint distribution of the number of
customers at both queues. The steady-state Laplace-Stieljes
transform and the mean of the sojourn time of the customers in the
system are also obtained.
We provide numerical examples and discuss the dependency of the
steady-state mean of the sojourn time of the customers on several
basic system parameters. Utilizing the structural characteristics of
the model we discuss the interpretation of the results. This gives
an insight into the behavior of this tandem queuing model and can be
a base for developing approximations for it.
Supervisory control for discrete event systems (DESs) belongs essentially
to the logic level for control problems in DESs. Its corresponding control task is hard. In this
paper, we study a new optimal control problem in DESs. The performance measure is to maximize
the maximal discounted total reward among all possible strings (i.e., paths) of the controlled
condition we need for this is only that the performance measure is well defined.
We then divide the problem into three sub-cases where the optimal values are respectively
finite, positive infinite and negative infinite. We then show the optimality equation in the
case with a finite optimal value. Also, we characterize the optimality equation together with
its solutions and characterize the structure of the set of all optimal policies. All the results
are still true when the performance measure is to maximize the minimal discounted total reward
among all possible strings of the controlled system.
Finally, we apply these equations and solutions to a resource allocation system. The system may
be deadlocked and in order to avoid the deadlock we can either prohibit occurrence
of some events or resolve the deadlock. It is shown that from the view of the maximal
discounted total cost, it is better to resolve the deadlock if and only if the cost for
resolving the deadlock is less than the threshold value.
Peer-to-Peer (P2P) storage systems are a prevalent and important
mode for implementing cost-efficient, large-scale distributed
storage. Considering the random departure feature of the peers and
the diverse popularity of the data objects, a proper number of
replicas needs to be maintained, and a reasonable trigger threshold
of replica repair needs to be set for high data availability and
low system overhead. In this paper, based on the working principle
of the lazy replica repair policy in a P2P storage system, a
three-dimensional Markov chain model is constructed, and the model
is analyzed in steady-state by using a matrix-geometric method.
Then, the performance measures in terms of the availability of one
data object, the average access latency, and the replication rate
are given. Moreover, numerical results with analysis are provided
to demonstrate how system parameters such as the replica number and
the replica repair instant influence the system performance.
Finally, we develop benefit functions to optimize the replica number
and the repair trigger threshold.
In this paper, we present a new model for optimal control of discrete event systems (DESs) with
an arbitrary control pattern. Here, a discrete event system is defined as a collection of event
sets that depend on strings. When the system generates a string, the next event that may occur
should be in the corresponding event set. In the optimal control model, there are rewards for
choosing control inputs at strings and the sets of available control inputs also depend on
strings. The performance measure is to find a policy under the condition where the discounted
total reward among strings from the initial state is maximized. By applying ideas from Markov
decision processes, we divide the problem into three sub-cases where the optimal value is
respectively finite, positive infinite and negative infinite. For the case with finite optimal
values, the optimality equation is shown and further characterized with its solutions. We also
characterize the structure of the set of all optimal policies. Moreover, we discuss invariance
and closeness of several languages. We present a new supervisory control problem of DESs with
the control pattern being dependent on strings. We study the problem in both the event feedback
control and the state feedback control by generalizing concepts of invariant and closed
languages/predicates. Finally, we apply the above model and results to a job-matching problem.
Supervisory control belongs essentially to the logic
level for control problems in discrete event systems (DESs) and its
corresponding control task is hard. This is unlike many practical optimal
control problems which belong to the performance level and whose control
tasks are soft. In this paper, we present two new optimal control problems
of DESs: one with cost functions for choosing control inputs, and the other
for occurring events. Their performance measures are to minimize the maximal
discounted total cost among all possible strings that the system generates.
Since this is a nonlinear optimization problem, we model such systems by
using Markov decision processes. We then present the optimality equations
for both control problems and obtain their optimal solutions. When the cost
functions are stationary, we show that both the optimality equations and
their solutions are also stationary. We then use these equations and
solutions to describe and solve uniformly the basic synthesizing problems in
the two branches of the supervisory control area: those being the event
feedback control and the state feedback control. Moreover, we show that the
control invariant languages and the control invariant predicates with their
permissive supervisors and state feedbacks not only have meanings in
supervisory control of DESs, but are also the optimal solutions for some
optimal control problems. This shows a link existing between the logic level
and the performance level for the control of discrete event systems.
Finally, a numerical example is given to illustrate some results for
supervisory control of a DES.
In this paper, we consider a cognitive radio network with multiple Secondary Users (SUs). The SU packets generated from the SUs are divided into SU1 packets and SU2 packets, and the SU1 packets have higher priority than the SU2 packets. Different from the conventional preemptive priority scheme (called Scheme Ⅰ), we propose a non-preemptive priority scheme for the SU1 packets (called Scheme Ⅱ) to guarantee the transmission continuity of the SU2 packets. By constructing a three-dimensional Markov chain, we give the transition probability matrix of the Markov chain, and obtain the steady-state distribution of the system model. Accordingly, we derive some performance measures, such as the channel utilization, the blocking probability of the SU1 packets, the interruption probability of the SU1 packets and the SU2 packets, the normalized throughput of the SU1 packets, and the average latency of the SU2 packets. Moreover, we provide numerical experiments to compare different performance measures between the two priority schemes. Finally, we show and compare the Nash equilibrium strategy and the socially optimal strategy for the SU2 packets between Scheme Ⅰ and Scheme Ⅱ.
This Special Issue of Numerical Algebra, Control and Optimization (NACO) is dedicated to Professor Yutaka Takahashi on the occasion of his 60th birthday and in recognition of his fundamental contributions in queueing theory and network applications. It is a great honor and a pleasure for the Guest Editors to have this privilege to edit this Special Issue.
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In this paper we present the analysis of an M/M/c multiple
synchronous vacation model. In contrast to the previous works on
synchronous vacation model we consider the model with gated service
discipline and with independent and identically distributed vacation
periods. The analysis of this model requires different methodology
compared to those ones used for synchronous vacation model so far.
We provide the probability-generating function and the mean of the
stationary number of customers at an arbitrary epoch as well as the
Laplace-Stieljes transform and the mean of the stationary waiting
time. The stationary distribution of the number of busy servers and
the stability of the system are also considered. In the final part
of the paper numerical examples illustrate the computational
This vacation queue is suitable to model a single operator
controlled system consisting of more machines. Hence the provided
analysis can be applied to study and optimize such systems.
In this paper, we study a two-server Markovian network system with
balking and a Bernoulli schedule under a single vacation policy,
where servers have different service rates. After every service,
only one server may take a vacation or continue to stay in the
system. The vacation time follows an exponential distribution. An
arriving customer finding both servers free will choose the faster
server. If the customer finds only one server is free, this customer
chooses this free server. If the customer finds both servers are not
free, then this customer may join the system or balk. For this
system, we obtain the steady state condition, the stationary
distribution of the number of customers in the system, and the mean
system size by using a matrix-geometric method. Some special cases
are deduced, which match with earlier exiting results. Extensive
numerical illustrations are provided. Motivation for this system
model also comes from some computer communication networks with
different types of traffic such as real-time traffic and
non-real-time traffic, where messages can be processed by two
channels (servers) with different transmission rates. The behavior
of abandoning messages can be equated with the balking of customers
in this system model.