5,495 research outputs found
Loss systems in a random environment
We consider a single server system with infinite waiting room in a random
environment. The service system and the environment interact in both
directions. Whenever the environment enters a prespecified subset of its state
space the service process is completely blocked: Service is interrupted and
newly arriving customers are lost. We prove an if-and-only-if-condition for a
product form steady state distribution of the joint queueing-environment
process. A consequence is a strong insensitivity property for such systems.
We discuss several applications, e.g. from inventory theory and reliability
theory, and show that our result extends and generalizes several theorems found
in the literature, e.g. of queueing-inventory processes.
We investigate further classical loss systems, where due to finite waiting
room loss of customers occurs. In connection with loss of customers due to
blocking by the environment and service interruptions new phenomena arise.
We further investigate the embedded Markov chains at departure epochs and
show that the behaviour of the embedded Markov chain is often considerably
different from that of the continuous time Markov process. This is different
from the behaviour of the standard M/G/1, where the steady state of the
embedded Markov chain and the continuous time process coincide.
For exponential queueing systems we show that there is a product form
equilibrium of the embedded Markov chain under rather general conditions. For
systems with non-exponential service times more restrictive constraints are
needed, which we prove by a counter example where the environment represents an
inventory attached to an M/D/1 queue. Such integrated queueing-inventory
systems are dealt with in the literature previously, and are revisited here in
detail
Correction. Brownian models of open processing networks: canonical representation of workload
Due to a printing error the above mentioned article [Annals of Applied
Probability 10 (2000) 75--103, doi:10.1214/aoap/1019737665] had numerous
equations appearing incorrectly in the print version of this paper. The entire
article follows as it should have appeared. IMS apologizes to the author and
the readers for this error. A recent paper by Harrison and Van Mieghem
explained in general mathematical terms how one forms an ``equivalent workload
formulation'' of a Brownian network model. Denoting by the state vector
of the original Brownian network, one has a lower dimensional state descriptor
in the equivalent workload formulation, where can be chosen as
any basis matrix for a particular linear space. This paper considers Brownian
models for a very general class of open processing networks, and in that
context develops a more extensive interpretation of the equivalent workload
formulation, thus extending earlier work by Laws on alternate routing problems.
A linear program called the static planning problem is introduced to articulate
the notion of ``heavy traffic'' for a general open network, and the dual of
that linear program is used to define a canonical choice of the basis matrix
. To be specific, rows of the canonical are alternative basic optimal
solutions of the dual linear program. If the network data satisfy a natural
monotonicity condition, the canonical matrix is shown to be nonnegative,
and another natural condition is identified which ensures that admits a
factorization related to the notion of resource pooling.Comment: Published at http://dx.doi.org/10.1214/105051606000000583 in the
Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute
of Mathematical Statistics (http://www.imstat.org
Strategies for a centralized single product multiclass M/G/1 make-to-stock queue
Make-to-stock queues are typically investigated in the M/M/1 settings. For centralized single-item systems with backlogs, the multilevel rationing (MR) policy is established as optimal and the strict priority (SP) policy is a practical compromise, balancing cost and ease of implementation. However, the optimal policy is unknown when service time is general, i.e., for M/G/1 queues. Dynamic programming, the tool commonly used to investigate the MR policy in make-to-stock queues, is less practical when service time is general. In this paper we focus on customer composition: the proportion of customers of each class to the total number of customers in the queue. We do so because the number of customers in M/G/1 queues is invariant for any nonidling and nonanticipating policy. To characterize customer composition, we consider a series of two-priority M/G/1 queues where the first service time in each busy period is different from standard service times, i.e., this first service time is exceptional. We characterize the required exceptional first service times and the exact solution of such queues. From our results, we derive the optimal cost and control for the MR and SP policies for M/G/1 make-to-stock queues
Queueing process with excluded-volume effect
We introduce an extension of the M/M/1 queueing process with a spatial
structure and excluded- volume effect. The rule of particle hopping is the same
as for the totally asymmetric simple exclusion process (TASEP). A
stationary-state solution is constructed in a slightly arranged matrix product
form of the open TASEP. We obtain the critical line that separates the
parameter space depending on whether the model has the stationary state. We
calculate the average length of the model and the number of particles and show
the monotonicity of the probability of the length in the stationary state. We
also consider a generalization of the model with backward hopping of particles
allowed and an alternate joined system of the M/M/1 queueing process and the
open TASEP.Comment: 9 figure
Dynamic Product Assembly and Inventory Control for Maximum Profit
We consider a manufacturing plant that purchases raw materials for product
assembly and then sells the final products to customers. There are M types of
raw materials and K types of products, and each product uses a certain subset
of raw materials for assembly. The plant operates in slotted time, and every
slot it makes decisions about re-stocking materials and pricing the existing
products in reaction to (possibly time-varying) material costs and consumer
demands. We develop a dynamic purchasing and pricing policy that yields time
average profit within epsilon of optimality, for any given epsilon>0, with a
worst case storage buffer requirement that is O(1/epsilon). The policy can be
implemented easily for large M, K, yields fast convergence times, and is robust
to non-ergodic system dynamics.Comment: 32 page
Restless bandit marginal productivity indices II: multiproject case and scheduling a multiclass make-to-order/-stock M/G/1 queue
This paper develops a framework based on convex optimization and economic ideas to formulate and solve approximately a rich class of dynamic and stochastic resource allocation problems, fitting in a generic discrete-state multi-project restless bandit problem (RBP). It draws on the single-project framework in the author's companion paper "Restless bandit marginal productivity indices I: Single-project case and optimal control of a make-to-stock M/G/1 queue", based on characterization of a project's marginal productivity index (MPI). Our framework significantly expands the scope of Whittle (1988)'s seminal approach to the RBP. Contributions include: (i) Formulation of a generic multi-project RBP, and algorithmic solution via single-project MPIs of a relaxed problem, giving a lower bound on optimal cost performance; (ii) a heuristic MPI-based hedging point and index policy; (iii) application of the MPI policy and bound to the problem of dynamic scheduling for a multiclass combined MTO/MTS M/G/1 queue with convex backorder and stock holding cost rates, under the LRA criterion; and (iv) results of a computational study on the MPI bound and policy, showing the latter's near-optimality across the cases investigated
The M/M/ N
This paper considers the M/M/N repairable queuing system. The customers' arrival
is a Poisson process. The servers are subject to breakdown according to Poisson processes with
different rates in idle time and busy time, respectively. The breakdown servers are repaired by repairmen,
and the repair time is an exponential distribution. Using probability generating function
and transform method, we obtain the steady-state probabilities of the system states, the steady-state
availability of the servers, and the mean queueing length of the model
Restless bandit marginal productivity indices I: singleproject case and optimal control of a make-to-stock M/G/1 queue
This paper develops a framework based on convex optimization and economic ideas to formulate and solve by an index policy the problem of optimal dynamic effort allocation to a generic discrete-state restless bandit (i.e. binary-action: work/rest) project, elucidating a host of issues raised by Whittle (1988)Žs seminal work on the topic. Our contributions include: (i) a unifying definition of a projectŽs marginal productivity index (MPI), characterizing optimal policies; (ii) a complete characterization of indexability (existence of the MPI) as satisfaction by the project of the law of diminishing returns (to effort); (iii) sufficient indexability conditions based on partial conservation laws (PCLs), extending previous results of the author from the finite to the countable state case; (iv) application to a semi-Markov project, including a new MPI for a mixed longrun-average (LRA)/ bias criterion, which exists in relevant queueing control models where the index proposed by Whittle (1988) does not; and (v) optimal MPI policies for service-controlled make-to-order (MTO) and make-to-stock (MTS) M/G/1 queues with convex back order and stock holding cost rates, under discounted and LRA criteria
- …