Infinite Orbit Research Articles

Modeling of Junior Servers Approaching a Senior Server in the Retrial Queuing-Inventory System

This article deals with the queuing-inventory system, composed of c junior servers, a senior server, two finite waiting halls, and an infinite orbit. On occasion, junior servers encounter challenges during customer service. In these instances, they approach the senior server for guidance in resolving the issue. Suppose the senior server is engaged with another junior server. The approaching junior servers await their turn in a finite waiting area with a capacity of c for consultation. Concerning this, we study the performance of junior servers approaching the senior server in the retrial queuing-inventory model with the two finite waiting halls dedicated to the primary customers and the junior servers for consultation. We formulate a level-dependent QBD process and solve its steady-state probability vector using Neuts and Rao’s truncation method. The stability condition of the system is derived and the R matrix is computed. The optimum total cost has been obtained, and the sensitivity analyses, which include the expected total cost, the waiting time of customers in the waiting hall and orbit, the number of busy servers, and a fraction of the successful retrial rate of the model, are computed numerically.

Similarity Classes of the Longest-Edge Trisection of Triangles

This paper studies the triangle similarity classes obtained by iterative application of the longest-edge trisection of triangles. The longest-edge trisection (3T-LE) of a triangle is obtained by joining the two points which divide the longest edge in three equal parts with the opposite vertex. This partition, as well as the longest-edge bisection (2T-LE), does not degenerate, which means that there is a positive lower bound to the minimum angle generated. However, unlike what happens with the 2T-LE, the number of similarity classes appearing by the iterative application of the 3T-LE to a single initial triangle is not finite in general. There are only three exceptions to this fact: the right triangle with its sides in the ratio 1:2:3 and the other two triangles in its orbit. This result, although of a combinatorial nature, is proved here with the machinery of discrete dynamics in a triangle shape space with hyperbolic metric. It is also shown that for a point with an infinite orbit, infinite points of the orbit are in three circles with centers at the points with finite orbits.

Weight modules over Bell–Rogalski algebras

We study a class of Z-graded algebras introduced by Bell and Rogalski. Their construction generalizes in large part that of rank one generalized Weyl algebras (GWAs). We establish certain ring-theoretic properties of these algebras and study their connection to GWAs. We classify the simple weight modules in the infinite orbit case and provide a partial classification in the case of orbits of finite order.

Development of a Robot for Decontamination of High Places and Decommissioning Work That Can Cope with Slopes and Steps

In this study, a robot was created to compete in the “7th Decommissioning Creative Robot Contest. The “Decommissioning Creative Robot Contest” is a robot contest in which the fields and tasks are set based on the assumption of decommissioning work at TEPCO’s Fukushima Daiichi Nuclear Power Station. The theme of this year’s competition is 3D decontamination of a highly contaminated area inside a nuclear reactor building. To solve the competition problem, we propose a robot for decommissioning work equipped with “an undercarriage mechanism that can cope with slopes and steps”, “a decontamination mechanism with two types of rotation trajectories”, and “a lifting mechanism with a high deformation rate. The undercarriage mechanism uses a four-wheel drive with infinite orbit to cope with slopes and steps. The decontamination mechanism uses two different types of orbits for decontamination: a circular rotating orbit and a longitudinal rotating orbit for rotation. For the lifting mechanism, three pantograph mechanisms were used. This study realized a control and mechanism to overcome steps and slopes, a deployment mechanism to high places, and a mechanism with two types of orbits to carefully decontaminate a large area.

A matrix-analytic method for the steady state analysis of a Markovian queueing system with scrap items

This paper studies a mathematical analysis of scrap item-based single server stochastic queueing, which consists of storage space for the scrap items collected from the customers. The primary customers arrive in the finite waiting hall to sell scrap or recyclable products. The server collects the customer’s scrap items and stores them in the storage space. If the waiting hall is full whenever the primary customers arrive, they can join an infinite orbit or permanently leave the system under the Bernoulli trial. Whenever the orbital customers find at least one free space in the waiting hall, they can enter it. The server begins vacation once the storage reaches its maximum capacity or no more customers exist in the waiting hall. The server switches to regular mode whenever at least one free space exists in the storage, and one customer is in the waiting hall. Otherwise, he is not working in vacation mode. Customers in the waiting hall may lose their patience after waiting for a long time and may either enter orbit or leave the system permanently. Whenever the storage level (scrap item level) reaches a certain limit, the server places the procurement order for sending the items to the recycling unit or re-manufacturer. The system’s steady-state probability vector and stability condition have been derived by truncating the orbit at a point. Various system performances, such as expected total cost, profit cost, and waiting time for customers in the waiting hall, are defined, and a numerical study is carried out to analyze the performance of the proposed model. The numerical findings indicate that disabling working in vacation mode increases waiting time and total cost while working in vacation mode results in minimized waiting time and total cost.

Analysis of Stochastic State-Dependent Arrivals in a Queueing-Inventory System with Multiple Server Vacation and Retrial Facility

This article analyses a four-dimensional stochastic queueing-inventory system with multiple server vacations and a state-dependent arrival process. The server can start multiple vacations at a random time only when there is no customer in the waiting hall and the inventory level is zero. The arrival flow of customers in the system is state-dependent. Whenever the arriving customer finds that the waiting hall is full, they enter into the infinite orbit and they retry to enter the waiting hall. If there is at least one space in the waiting hall, the orbital customer enters the waiting hall. When the server is on vacation, the primary (retrial) customer enters the system with a rate of λ1(θ1). If the server is not on vacation, the primary (retrial) arrival occurs with a rate of λ2(θ2). Each arrival rate follows an independent Poisson distribution. The service is provided to customers one by one in a positive time with the rate of μ, which follows exponential distribution. When the inventory level drops to a fixed s, reorder of Q items is triggered immediately under (s,Q) ordering policy. The stability of the system has been analysed, and using the Neuts matrix geometric approach, the stationary probability vectors have been obtained. Moreover, various system performance measures are derived. The expected total cost analysis explores and verifies the characteristics of the assumed parameters of this model. The average waiting time of a customer in the waiting hall and orbit are investigated using all the parameters. The monotonicity of the parameters is verified with its characteristics by the numerical simulation. The discussion about the fraction time server being on vacation suggests that as the server’s vacation duration reduces, its fraction time also reduces. The mean number of customers in the waiting hall and orbit is reduced whenever the average service time per customer and average replenishment time are reduced.

Modified Erlang Loss System for Cognitive Wireless Networks

This paper considers a modified Erlang loss system for cognitive wireless networks and related applications. A primary user has pre-emptive priority over secondary users, and the primary customer is lost if upon arrival all the channels are used by other primary users. Secondary users cognitively use idle channels, and they can stay (either in an infinite buffer or in an orbit) in cases where idle channels are not available upon arrival or they are interrupted by primary users. While the infinite buffer model represents the case with zero sensing time, the infinite orbit model represents the case with positive sensing time. We obtain an explicit stability condition for the cases where arrival processes of primary users and secondary users follow Poisson processes, and their service times follow two distinct arbitrary distributions. The stability condition is insensitive to the service time distributions and implies the maximal throughout of secondary users. Moreover, we extend the stability analysis to the system with outgoing calls. For a special case of exponential service time distributions, we analyze the buffered system in depth to show the effect of parameters on the delay performance and the mean number of interruptions of secondary users. Our simulations for distributions rather than exponential reveal that the mean number of terminations for secondary users is less sensitive to the service time distribution of primary users.

Analysis of Stock-Dependent Arrival Process in a Retrial Stochastic Inventory System with Server Vacation

The present study deals with the stock-dependent Markovian demand of a retrial queueing system with a single server and multiple server vacation. The items are restocked under a continuous review (s,Q) ordering policy. When there is no item in the system, the server goes on vacation. Further, any arrival demand permits entry into an infinite orbit whenever the server is on vacation. In the Matrix geometric approach with the Neuts-Rao truncation technique, the steady-state joint distribution of the number of customers in orbit, the server status, and the inventory level is obtained. Under the steady-state conditions, some significant system performance measures, including the long-run total cost rate, are derived, and the Laplace-Stieltjes transform is also used to investigate the waiting time distribution. According to various considerations of uncontrollable parameters and costs, the merits of the proposed model, especially the important characteristics of the system with stock dependency over non-stock dependency, are explored. Ultimately, the important facts and ideas behind this model are given in conclusion.

Retrial BMAP/PH/N Queueing System with a Threshold-Dependent Inter-Retrial Time Distribution

In this paper, we study a multi-server queueing system with retrials and an infinite orbit. The arrival of primary customers is described by a batch Markovian arrival process (BMAP), and the service times have a phase-type (PH) distribution. Previously, in the literature, such a system was mainly considered under the strict assumption that the intervals between the repeated attempts from the orbit have an exponential distribution. Only a few publications dealt with retrial queueing systems with non-exponential inter-retrial times. These publications assumed either the rate of retrials is constant regardless of the number of customers in the orbit or this rate is constant when the number of orbital customers exceeds a certain threshold. Such assumptions essentially simplify the mathematical analysis of the system, but do not reflect the nature of the majority of real-life retrial processes. The main feature of the model under study is that we considered the classical retrial strategy under which the retrial rate is proportional to the number of orbital customers. However, in this case, the assumption of the non-exponential distribution of inter-retrial times leads to insurmountable computational difficulties. To overcome these difficulties, we supposed that inter-retrial times have a phase-type distribution if the number of customers in the orbit is less than or equal to some non-negative integer (threshold) and have an exponential distribution in the contrary case. By appropriately choosing the threshold, one can obtain a sufficiently accurate approximation of the system with a PH distribution of the inter-retrial times. Thus, the model under study takes into account the realistic nature of the retrial process and, at the same time, does not resort to restrictions such as a constant retrial rate or to rough truncation methods often applied to the analysis of retrial queueing systems with an infinite orbit. We describe the behavior of the system by a multi-dimensional Markov chain, derive the stability condition, and calculate the steady-state distribution and the main performance indicators of the system. We made sure numerically that there was a reasonable value of the threshold under which our model can be served as a good approximation of the BMAP/PH/N queueing system with the PH distribution of inter-retrial times. We also numerically compared the system under consideration with the corresponding queueing system having exponentially distributed inter-retrial times and saw that the latter is a poor approximation of the system with the PH distribution of inter-retrial times. We present a number of illustrative numerical examples to analyze the behavior of the system performance indicators depending on the system parameters, the variance of inter-retrial times, and the correlation in the input flow.

On the orbits of automaton semigroups and groups

We investigate the orbits of automaton semigroups and groups to obtain algorithmic and structural results, both for general automata but also for some special subclasses. First, we show that a more general version of the finiteness problem for automaton groups is undecidable. This problem is equivalent to the finiteness problem for left principal ideals in automaton semigroups generated by complete and reversible automata. Then, we look at w-word (i.\,e.\ right infinite words) with a finite orbit. We show that every automaton yielding an w-word with a finite orbit already yields an ultimately periodic one, which is not periodic in general, however. On the algorithmic side, we observe that it is not possible to decide whether a given periodic w-word has an infinite orbit and that we cannot check whether a given reversible and complete automaton admits an w-word with a finite orbit, a reciprocal problem to the finiteness problem for automaton semigroups in the reversible case. Finally, we look at automaton groups generated by reversible but not bi-reversible automata and show that many words have infinite orbits under the action of such automata.

Permutable quasiregular maps

AbstractLet f and g be two quasiregular maps in $\mathbb{R}^d$ that are of transcendental type and also satisfy $f\circ g =g \circ f$ . We show that if the fast escaping sets of those functions are contained in their respective Julia sets then those two functions must have the same Julia set. We also obtain the same conclusion about commuting quasimeromorphic functions with infinite backward orbit of infinity. Furthermore we show that permutable quasiregular functions of the form f and $g = \phi \circ f$ , where $\phi$ is a quasiconformal map, have the same Julia sets and that polynomial type quasiregular maps cannot commute with transcendental type ones unless their degree is less than or equal to their dilatation.

Counting walks with large steps in an orthant

In the past fifteen years, the enumeration of lattice walks with steps taken in a prescribed set \mathcal S and confined to a given cone, especially the first quadrant of the plane, has been intensely studied. As a result, the generating functions of quadrant walks are now well-understood, provided the allowed steps are small , that is \mathcal S \subset \{-1, 0,1\}^2 . In particular, having small steps is crucial for the definition of a certain group of bi-rational transformations of the plane. It has been proved that this group is finite if and only if the corresponding generating function is D-finite (that is, it satisfies a linear differential equation with polynomial coefficients). This group is also the key to the uniform solution of 19 of the 23 small step models possessing a finite group. In contrast, almost nothing is known for walks with arbitrary steps. In this paper, we extend the definition of the group, or rather of the associated orbit, to this general case, and generalize the above uniform solution of small step models. When this approach works, it invariably yields a D-finite generating function. We apply it to many quadrant problems, including some infinite families. After developing the general theory, we consider the 13110 two-dimensional models with steps in \{-2,-1,0,1\}^2 having at least one -2 coordinate. We prove that only 240 of them have a finite orbit, and solve 231 of them with our method. The nine remaining models are the counterparts of the four models of the small step case that resist the uniform solution method (and which are known to have an algebraic generating function). We conjecture D-finiteness for their generating functions, but only two of them are likely to be algebraic. We also prove non-D-finiteness for the 12870 models with an infinite orbit, except for 16 of them.

Analysis of Interconnected Arrivals on Queueing-Inventory System with Two Multi-Server Service Channels and One Retrial Facility

Present-day queuing inventory systems (QIS) do not utilize two multi-server service channels. We proposed two multi-server service channels referred to as T1S (Type 1 n-identical multi-server) and T2S (Type 2 m-identical multi-server). It includes an optional interconnected service connection between T1S and T2S, which has a finite queue of size N. An arriving customer either uses the inventory (basic service or main service) for their demand, whom we call T1, or simply uses the service only, whom we call T2. Customer T1 will utilize the server T1S, while customer T2 will utilize the server T2S, and T1 can also get the second optional service after completing their main service. If there is a free server with a positive inventory, there is a chance that T1 customers may go to an infinite orbit whenever they find that either all the servers are busy or no sufficient stock. The orbital customer can request for T1S service under the classical retrial policy. Q(=S−s) items are replaced into the inventory whenever it falls into the reorder level s such that the inequality always holds n<s. We use the standard (s,Q) ordering policy to replace items into the inventory. By varying S and s, we investigate to find the optimal cost value using stationary probability vector ϕ. We used the Neuts Matrix geometric approach to derive the stability condition and steady-state analysis with R-matrix to find ϕ. Then, we perform the waiting time analysis for both T1 and T2 customers using Laplace transform technique. Further, we computed the necessary system characteristics and presented sufficient numerical results.

Stochastic modeling on M/M/1/N inventory system with queue-dependent service rate and retrial facility

This paper investigates the queue-dependent service rates(QDSR) in the stochastic queueing-inventory system(SQIS). This SQIS consists a single server service channel, $ S $ number of inventories, and a finite queue. An arriving customer gets the service immediately if the server is free and there exists a positive stock in the SQIS. When the server is busy, they have to wait in the finite queue. Suppose they find that the waiting hall is full, either they leave the system or enter into an infinite orbit under the Bernoulli trial. The service rate of any arrival is dependent on the number of customers in the queue at present. The orbital customer can compete for the service only through joining into the waiting hall based on the classical retrial policy. Whenever the number of inventories in SQIS is reached a certain limit $ s $, the replenishment of $ Q( = S-s) $ items is placed. Due to the structure of rate matrix, the stability analysis, minimal non-negative solutions of the quadratic systems are derived through the Neuts matrix-geometric approximation(MGA). Further, the waiting time distribution(WTD) of arrival and necessary system characteristics are derived. Finally, adequate numerical examples are presented to highlight the proposed SQIS.

A MAP/PH/1 Perishable Inventory System with Dependent Retrial Loss

This paper considers a continuous review perishable inventory system with an infinite orbit. The replenishment of items in the inventory is governed by (s, Q) policy. The Markovian Arrival Process characterizes the arrival pattern of primary customers and the service time follows phase-type distribution. Inventoried items are subject to decay with some preassigned linear rate. The items are purchased from an outside source and lead time is exponentially distributed. An unsatisfied primary customer may enter an orbit or exit the system with a predetermined probability. The retrial customers, those who entered into the orbit, try to access the idle server with a linear rate. An unsatisfied retrial customer may enter or exit the orbit with prefixed probability based on the inventory level. Stationary probability vectors are obtained using the Matrix Analytic method. Numerical optimization of the profit function is carried out for different parameter values.

Infinite automaton semigroups and groups have infinite orbits

We show that an automaton group or semigroup is infinite if and only if it admits an ω-word (i.e. a right-infinite word) with an infinite orbit, which solves an open problem communicated to us by I. V. Bondarenko. In fact, we prove a generalization of this result, which can be applied to show that finitely generated subgroups and subsemigroups as well as principal left ideals of automaton semigroups are infinite if and only if there is an ω-word with an infinite orbit under their action. The proof also shows some interesting connections between the automaton semigroup and its dual. Finally, our result is interesting from an algorithmic perspective as it allows for a re-formulation of the finiteness problem for automaton groups and semigroups.

Infinite orbit depth and length of Melnikov functions

In this paper we study polynomial Hamiltonian systems dF=0 in the plane and their small perturbations: dF+ϵω=0. The first nonzero Melnikov function Mμ=Mμ(F,γ,ω) of the Poincaré map along a loop γ of dF=0 is given by an iterated integral [3]. In [7], we bounded the length of the iterated integral Mμ by a geometric number k=k(F,γ) which we call orbit depth. We conjectured that the bound is optimal.Here, we give a simple example of a Hamiltonian system F and its orbit γ having infinite orbit depth. If our conjecture is true, for this example there should exist deformations dF+ϵω with arbitrary high length first nonzero Melnikov function Mμ along γ. We construct deformations dF+ϵω=0 whose first nonzero Melnikov function Mμ is of length three and explain the difficulties in constructing deformations having high length first nonzero Melnikov functions Mμ.

Infinite distances in field space and massless towers of states

It has been conjectured that in theories consistent with quantum gravity infinite distances in field space coincide with an infinite tower of states becoming massless exponentially fast in the proper field distance. The complex-structure moduli space of Calabi-Yau manifolds is a good testing ground for this conjecture since it is known to encode quantum gravity physics. We study infinite distances in this setting and present new evidence for the above conjecture. Points in moduli space which are at infinite proper distance along any path are characterised by an infinite order monodromy matrix. We utilise the nilpotent orbit theorem to show that for a large class of such points the monodromy matrix generates an infinite orbit within the spectrum of BPS states. We identify an infinite tower of states with this orbit. Further, the theorem gives the local metric on the moduli space which can be used to show that the mass of the states decreases exponentially fast upon approaching the point. We also propose a reason for why infinite distances are related to infinite towers of states. Specifically, we present evidence that the infinite distance itself is an emergent quantum phenomenon induced by integrating out at one-loop the states that become massless. Concretely, we show that the behaviour of the field space metric upon approaching infinite distance can be recovered from integrating out the BPS states. Similarly, at infinite distance the gauge couplings of closed-string Abelian gauge symmetries vanish in a way which can be matched onto integrating out the infinite tower of charged BPS states. This presents evidence towards the idea that also the gauge theory weak-coupling limit can be thought of as emergent.

A retrial inventory system with an unreliable server

In this paper, we analyse an unreliable (s, S) inventory system with infinite capacity orbit. Customers join the retrial orbit if they are interrupted by a server breakdown and repeatedly attempt to access the server at iid intervals until it is found functioning and idle. We provide stability condition and steady state system size probabilities of the underlying stochastic process. Also, we provide several other characteristics of the system like average waiting time of a customer in the orbit, average number of breakdowns of the server, average duration of a failed service, etc. We obtained a total expected cost function which is optimal in both s and S and provided some numerical illustration.

Minimality and the Rauzy – Veech algorithm for interval exchange transformations with flips

Oriented interval exchange transformations (i.e.t.s) admitting infinite orbit of the Rauzy–Veech operator are minimal. In this paper, we analyse what happens for non-oriented i.e.t.s. In particular, we prove an analogous result for i.e.t.s with flips (Theorem A). We also show that the behaviour of i.e.t.s with flips is different to the one of oriented i.e.t.s, see Proposition 11.