Michael Zazanis - Academia.edu (original) (raw)

Papers by Michael Zazanis

Operations Research, 1992

In Poland, there has established a new sub-discipline of social science that is being labeled as ... more In Poland, there has established a new sub-discipline of social science that is being labeled as counselogy and created by Alicja Kargulowa. The main research object of counselogy is counselling, broadly understood, and therefore, counselogy deals with the analysis of psychological counselling, career counselling, family counselling, marriage counselling, education counselling, bank counseling, etc.. This text contains information on establishing the elitist Counselogical Association. There are the counselogists and counselling researchers among the members of this association. A regular member may be a physical person holding an academic rank of at least Ph.D.. The mission of this association is research activity: initiating research projects, conducting international cooperation with academic centres, integrating counselogy experts, popularizing knowledge of counselogy by organizing conferences and emitting publications and cooperation with practitioners to introduce innovations into counselling.

In this paper we examine a reinsurance market where a number of companies co-operate in order to ... more In this paper we examine a reinsurance market where a number of companies co-operate in order to minimize the total premium. The analysis is carried out in the context of the classical Cramér–Lundberg model of collective risk theory and it is assumed that each company determines its premium based on its attitude towards risk which is expressed via a fixed, infinite horizon ruin probability, as specified by the model. We formulate this problem as a variational problem in which ruin probabilities (as represented by the corresponding adjustment coefficients) are treated as constraints and the sum of the premium rates charged by insurer and reinsurers is minimized. Within this framework, the optimal solution is proportional reinsurance. Connections are also made with classical studies of the reinsurance market, such as Borch (1962).

ArXiv, 2021

When designing a message transmission system, from the point of view of making sure that the info... more When designing a message transmission system, from the point of view of making sure that the information transmitted is as fresh as possible, two rules of thumb seem reasonable: use small buffers and adopt a last-in-first-out policy. In this paper, we measure freshness of information using the “age of information” performance measure. Considering it as a stochastic process operating in a stationary regime, we compute not just the first moment but the whole marginal distribution of the age of information (something important in applications) for two well-performing systems. In neither case do we allow for preemption of the message being processed because this may be difficult to implement in practice. We assume that the arrival process is Poisson and that the messages have independent sizes (service times) with common distribution. We use Palm and Markov-renewal theory to derive explicit results for Laplace transforms. In particular, this approach can be used to analyze more complex ...

© Applied Probability Trust 2018. Stirling numbers of the second kind, S(n, r), denote the number... more © Applied Probability Trust 2018. Stirling numbers of the second kind, S(n, r), denote the number of partitions of a finite set of size n into r disjoint nonempty subsets. The aim of this short article is to shed some light on the generating functions of these numbers by deriving them probabilistically. We do this by linking them to Markov chains related to the classical coupon collector problem; coupons are collected in discrete time (ordinary generating function) or in continuous time (exponential generating function). We also review the shortest possible combinatorial derivations of these generating functions.

We consider a queueing system with batch Poisson arrivals and Bernoulli feedback which is subject... more We consider a queueing system with batch Poisson arrivals and Bernoulli feedback which is subject to disasters occurring according to an independent Poisson process. Each disaster is followed by a repair period. Service, vacation, and repair times are distributed according to general distributions. We analyze this system using the supplementary variables technique and we obtain the probability generating function of the stationary queue length distribution, the Laplace transform of the busy period distribution, and the probability that a customer completes service. The Laplace transform of the time between two consecutive disasters is also obtained. These results are used to obtain the Laplace transform of the system time distribution for a typical customer who completes service and departs from the system without being affected by a disaster. Finally, we consider a variant of the system where disasters can occur even when the server is not busy.

We examine Push and Pull production control systems under a make-toorder policy with safety stock... more We examine Push and Pull production control systems under a make-toorder policy with safety stock. We compute the distribution of the waiting time until a demand is satisfied, as well as flow time distributions and service levels for both systems. Work-in-process levels are determined as well. The analysis is carried out under Markovian assumptions and the explicit results on flow times depend on non-overtake conditions that are satisfied for single machine stations.

ArXiv, 2021

Consider a message processing system whose objective is to produce the most current information a... more Consider a message processing system whose objective is to produce the most current information as measured by the quantity known as “age of information”. We have argued in previous papers that if we are allowed to design the message processing policy ad libitum, we should keep a small buffer and operate according to a LIFO policy. In this small note we provide an analysis for the AoI of the Pm system which uses a buffer of size m, a single server, operating without service preemption and in a LIFO manner for stored messages. Analytical expressions for the mean (or even distribution) of the AoI in steady-state are possible but with the aid computer algebra. We explain the the analysis for m = 3. 1 The background Here is the definition of the Age-of-Information (AoI) stochastic process α(t), t ∈ R, for a fairly general message processing system. Messages arrive at the times of a point process. A message may be rejected upon arrival or at any point during its sojourn in the system. If...

arXiv: Probability, 2016

This paper presents a stochastic approach to theorems concerning the behavior of iterations of th... more This paper presents a stochastic approach to theorems concerning the behavior of iterations of the Bernstein operator Bn taking a continuous function f ∈ C[0,1] to a degree-n polynomial when the number of iterations k tends to infinity and n is kept fixed or when n tends to infinity as well. In the first instance, the underlying stochastic process is the so-called Wright-Fisher model, whereas, in the second instance, the underlying stochastic process is the Wright-Fisher diffusion. Both processes are probably the most basic ones in mathematical genetics. By using Markov chain theory and stochastic compositions, we explain probabilistically a theorem due to Kelisky and Rivlin, and by using stochastic calculus we compute a formula for the application of Bn a number of times k = k(n) to a polynomial f when k(n)/n tends to a constant.

arXiv: Performance, 2019

A queueing system handling a sequence of message arrivals is considered where each message obsole... more A queueing system handling a sequence of message arrivals is considered where each message obsoletes all previous messages. The objective is to assess the freshness of the latest message/information that has been successfully transmitted, i.e., "age of information" (AoI). We study a variation of traditional AoI, the "Relative AoI", here defined so as to account for the presence of newly arrived messages/information to the queue to be transmitted.

ArXiv, 2021

Age of Information (AoI) has emerged as an important quality-of-service measure for applications ... more Age of Information (AoI) has emerged as an important quality-of-service measure for applications that prioritize delivery of the freshest information, e.g., virtual or augmented reality over mobile devices and wireless sensor networks used in the control of cyber-physical systems. We derive the Laplace transform of the stationary AoI for the M/GI/1/2 system with a “dynamic” service preemption and pushout policy depending on the existing service time of the in-service message. Thus, our system generalizes both the static M/GI/1/2 queue-pushout system without service preemption and the M/GI/1/1 bufferless system with service preemption two systems considered to provide very good AoI performance. Based on our analysis, for a service-time distribution that is a mixture of deterministic and exponential, we numerically show that the dynamic policy has lower mean AoI than that of these two static policies and also that of the well studied M/GI/1/1 blocking system.

We consider a cyclic closed-queueing network with arbitrary service time distributions and derive... more We consider a cyclic closed-queueing network with arbitrary service time distributions and derive firstand secondderivative estimators of some finite horizon performance metrics with respect to a parameter of any one of the service distributions. Our approach is based on observing a single sample path of this system and evaluating firstand second-order effects on departure times as a result of the parameter perturbation. We then define an estimator as a conditional expectation over appropriate observable quantities, using smoothed perturbation analysis techniques. This process recovers the first-derivative estimator along the way and gives new insights into event order change phenomena which are of higher order. Despite the complexity of the analysis, the final algorithms we obtain are relatively simple. Further, we show that our estimators are unbiased and include some numerical examples. We also show the use of our estimators in obtaining approximations of the entire system respon...

Queueing Systems, 2020

The idea behind the recently introduced "age of information" performance measure of a networked m... more The idea behind the recently introduced "age of information" performance measure of a networked message processing system is that it indicates our knowledge regarding the "freshness" of the most recent piece of information that can be used as a criterion for real-time control. In this foundational paper, we examine two such measures, one that has been extensively studied in the recent literature and a new one that could be more relevant from the point of view of the processor. Considering these measures as stochastic processes in a stationary environment (defined by the arrival processes, message processing times and admission controls in bufferless systems), we characterize their distributions using the Palm inversion formula. Under renewal assumptions we derive explicit solutions for their Laplace transforms and show some interesting decomposition properties. Previous work has mostly focused on computation of expectations in very particular cases. We argue that using bufferless or very small buffer systems is best

Expositiones Mathematicae, 2018

Annals of Operations Research, 2015

Journal of Applied Probability, 1995

Naval Research Logistics (NRL), 2015

We consider a queueing system with batch Poisson arrivals subject to disasters which occur only w... more We consider a queueing system with batch Poisson arrivals subject to disasters which occur only when the server is busy and clear the system. Following a disaster the server initiates a repair period during which arriving customers accumulate without receiving service. The server operates under a Multiple Adapted Vacation policy. We analyze this system using the supplementary variables technique and obtain the probability generating function of the number of customers in the system in stationarity the fraction of customers who complete service, and the Laplace transform of the system time of a typical customer in stationarity. Finally, we examine a variation of the model in which the system is subject to disasters even when the server is taking a vacation or is under repair.

Proceedings of the Fourth International Conference on Computer Integrated Manufacturing and Automation Technology

We examine Push and Pull production control systems under a make-to-order policy with safety stoc... more We examine Push and Pull production control systems under a make-to-order policy with safety stock. We compute the distribution of the waiting time until a demand is satisfied, as well as flow time distributions and service levels for both systems. Work-in-process levels are determined as well. The analysis is carried out under Markovian assumptions and the explicit results on flow times depend on non-overtake conditions that are satisfied for single machine stations.

Operations Research, 1992

In Poland, there has established a new sub-discipline of social science that is being labeled as ... more In Poland, there has established a new sub-discipline of social science that is being labeled as counselogy and created by Alicja Kargulowa. The main research object of counselogy is counselling, broadly understood, and therefore, counselogy deals with the analysis of psychological counselling, career counselling, family counselling, marriage counselling, education counselling, bank counseling, etc.. This text contains information on establishing the elitist Counselogical Association. There are the counselogists and counselling researchers among the members of this association. A regular member may be a physical person holding an academic rank of at least Ph.D.. The mission of this association is research activity: initiating research projects, conducting international cooperation with academic centres, integrating counselogy experts, popularizing knowledge of counselogy by organizing conferences and emitting publications and cooperation with practitioners to introduce innovations into counselling.

In this paper we examine a reinsurance market where a number of companies co-operate in order to ... more In this paper we examine a reinsurance market where a number of companies co-operate in order to minimize the total premium. The analysis is carried out in the context of the classical Cramér–Lundberg model of collective risk theory and it is assumed that each company determines its premium based on its attitude towards risk which is expressed via a fixed, infinite horizon ruin probability, as specified by the model. We formulate this problem as a variational problem in which ruin probabilities (as represented by the corresponding adjustment coefficients) are treated as constraints and the sum of the premium rates charged by insurer and reinsurers is minimized. Within this framework, the optimal solution is proportional reinsurance. Connections are also made with classical studies of the reinsurance market, such as Borch (1962).

ArXiv, 2021

When designing a message transmission system, from the point of view of making sure that the info... more When designing a message transmission system, from the point of view of making sure that the information transmitted is as fresh as possible, two rules of thumb seem reasonable: use small buffers and adopt a last-in-first-out policy. In this paper, we measure freshness of information using the “age of information” performance measure. Considering it as a stochastic process operating in a stationary regime, we compute not just the first moment but the whole marginal distribution of the age of information (something important in applications) for two well-performing systems. In neither case do we allow for preemption of the message being processed because this may be difficult to implement in practice. We assume that the arrival process is Poisson and that the messages have independent sizes (service times) with common distribution. We use Palm and Markov-renewal theory to derive explicit results for Laplace transforms. In particular, this approach can be used to analyze more complex ...

© Applied Probability Trust 2018. Stirling numbers of the second kind, S(n, r), denote the number... more © Applied Probability Trust 2018. Stirling numbers of the second kind, S(n, r), denote the number of partitions of a finite set of size n into r disjoint nonempty subsets. The aim of this short article is to shed some light on the generating functions of these numbers by deriving them probabilistically. We do this by linking them to Markov chains related to the classical coupon collector problem; coupons are collected in discrete time (ordinary generating function) or in continuous time (exponential generating function). We also review the shortest possible combinatorial derivations of these generating functions.

We consider a queueing system with batch Poisson arrivals and Bernoulli feedback which is subject... more We consider a queueing system with batch Poisson arrivals and Bernoulli feedback which is subject to disasters occurring according to an independent Poisson process. Each disaster is followed by a repair period. Service, vacation, and repair times are distributed according to general distributions. We analyze this system using the supplementary variables technique and we obtain the probability generating function of the stationary queue length distribution, the Laplace transform of the busy period distribution, and the probability that a customer completes service. The Laplace transform of the time between two consecutive disasters is also obtained. These results are used to obtain the Laplace transform of the system time distribution for a typical customer who completes service and departs from the system without being affected by a disaster. Finally, we consider a variant of the system where disasters can occur even when the server is not busy.

We examine Push and Pull production control systems under a make-toorder policy with safety stock... more We examine Push and Pull production control systems under a make-toorder policy with safety stock. We compute the distribution of the waiting time until a demand is satisfied, as well as flow time distributions and service levels for both systems. Work-in-process levels are determined as well. The analysis is carried out under Markovian assumptions and the explicit results on flow times depend on non-overtake conditions that are satisfied for single machine stations.

ArXiv, 2021

Consider a message processing system whose objective is to produce the most current information a... more Consider a message processing system whose objective is to produce the most current information as measured by the quantity known as “age of information”. We have argued in previous papers that if we are allowed to design the message processing policy ad libitum, we should keep a small buffer and operate according to a LIFO policy. In this small note we provide an analysis for the AoI of the Pm system which uses a buffer of size m, a single server, operating without service preemption and in a LIFO manner for stored messages. Analytical expressions for the mean (or even distribution) of the AoI in steady-state are possible but with the aid computer algebra. We explain the the analysis for m = 3. 1 The background Here is the definition of the Age-of-Information (AoI) stochastic process α(t), t ∈ R, for a fairly general message processing system. Messages arrive at the times of a point process. A message may be rejected upon arrival or at any point during its sojourn in the system. If...

arXiv: Probability, 2016

This paper presents a stochastic approach to theorems concerning the behavior of iterations of th... more This paper presents a stochastic approach to theorems concerning the behavior of iterations of the Bernstein operator Bn taking a continuous function f ∈ C[0,1] to a degree-n polynomial when the number of iterations k tends to infinity and n is kept fixed or when n tends to infinity as well. In the first instance, the underlying stochastic process is the so-called Wright-Fisher model, whereas, in the second instance, the underlying stochastic process is the Wright-Fisher diffusion. Both processes are probably the most basic ones in mathematical genetics. By using Markov chain theory and stochastic compositions, we explain probabilistically a theorem due to Kelisky and Rivlin, and by using stochastic calculus we compute a formula for the application of Bn a number of times k = k(n) to a polynomial f when k(n)/n tends to a constant.

arXiv: Performance, 2019

A queueing system handling a sequence of message arrivals is considered where each message obsole... more A queueing system handling a sequence of message arrivals is considered where each message obsoletes all previous messages. The objective is to assess the freshness of the latest message/information that has been successfully transmitted, i.e., "age of information" (AoI). We study a variation of traditional AoI, the "Relative AoI", here defined so as to account for the presence of newly arrived messages/information to the queue to be transmitted.

ArXiv, 2021

Age of Information (AoI) has emerged as an important quality-of-service measure for applications ... more Age of Information (AoI) has emerged as an important quality-of-service measure for applications that prioritize delivery of the freshest information, e.g., virtual or augmented reality over mobile devices and wireless sensor networks used in the control of cyber-physical systems. We derive the Laplace transform of the stationary AoI for the M/GI/1/2 system with a “dynamic” service preemption and pushout policy depending on the existing service time of the in-service message. Thus, our system generalizes both the static M/GI/1/2 queue-pushout system without service preemption and the M/GI/1/1 bufferless system with service preemption two systems considered to provide very good AoI performance. Based on our analysis, for a service-time distribution that is a mixture of deterministic and exponential, we numerically show that the dynamic policy has lower mean AoI than that of these two static policies and also that of the well studied M/GI/1/1 blocking system.

We consider a cyclic closed-queueing network with arbitrary service time distributions and derive... more We consider a cyclic closed-queueing network with arbitrary service time distributions and derive firstand secondderivative estimators of some finite horizon performance metrics with respect to a parameter of any one of the service distributions. Our approach is based on observing a single sample path of this system and evaluating firstand second-order effects on departure times as a result of the parameter perturbation. We then define an estimator as a conditional expectation over appropriate observable quantities, using smoothed perturbation analysis techniques. This process recovers the first-derivative estimator along the way and gives new insights into event order change phenomena which are of higher order. Despite the complexity of the analysis, the final algorithms we obtain are relatively simple. Further, we show that our estimators are unbiased and include some numerical examples. We also show the use of our estimators in obtaining approximations of the entire system respon...

Queueing Systems, 2020

The idea behind the recently introduced "age of information" performance measure of a networked m... more The idea behind the recently introduced "age of information" performance measure of a networked message processing system is that it indicates our knowledge regarding the "freshness" of the most recent piece of information that can be used as a criterion for real-time control. In this foundational paper, we examine two such measures, one that has been extensively studied in the recent literature and a new one that could be more relevant from the point of view of the processor. Considering these measures as stochastic processes in a stationary environment (defined by the arrival processes, message processing times and admission controls in bufferless systems), we characterize their distributions using the Palm inversion formula. Under renewal assumptions we derive explicit solutions for their Laplace transforms and show some interesting decomposition properties. Previous work has mostly focused on computation of expectations in very particular cases. We argue that using bufferless or very small buffer systems is best

Expositiones Mathematicae, 2018

Annals of Operations Research, 2015

Journal of Applied Probability, 1995

Naval Research Logistics (NRL), 2015

We consider a queueing system with batch Poisson arrivals subject to disasters which occur only w... more We consider a queueing system with batch Poisson arrivals subject to disasters which occur only when the server is busy and clear the system. Following a disaster the server initiates a repair period during which arriving customers accumulate without receiving service. The server operates under a Multiple Adapted Vacation policy. We analyze this system using the supplementary variables technique and obtain the probability generating function of the number of customers in the system in stationarity the fraction of customers who complete service, and the Laplace transform of the system time of a typical customer in stationarity. Finally, we examine a variation of the model in which the system is subject to disasters even when the server is taking a vacation or is under repair.

Proceedings of the Fourth International Conference on Computer Integrated Manufacturing and Automation Technology

We examine Push and Pull production control systems under a make-to-order policy with safety stoc... more We examine Push and Pull production control systems under a make-to-order policy with safety stock. We compute the distribution of the waiting time until a demand is satisfied, as well as flow time distributions and service levels for both systems. Work-in-process levels are determined as well. The analysis is carried out under Markovian assumptions and the explicit results on flow times depend on non-overtake conditions that are satisfied for single machine stations.