We consider a simple discrete time controlled queueing system, where the controller has a choice ... more We consider a simple discrete time controlled queueing system, where the controller has a choice of which server to use at each time slot and server performance varies according to a Markov modulated random environment. We explore the role of information in the system *
We consider queueing output processes of some elementary queueing models such as the M/M/1/K queu... more We consider queueing output processes of some elementary queueing models such as the M/M/1/K queue and the M/G/1 queue. An important performance measure for these counting processes is their variance curve, indicating the variance of the number of served customers over a time interval. Recent work has revealed some non-trivial properties dealing with the asymptotic rate at which the variance curve grows. In this paper we add to the results by finding explicit expressions for the second order approximation of the variance curve, namely the y-intercept of the linear asymptote. For M/M/1/K queues our results are based on the Drazin inverse of the generator. It turns out that by viewing output processes as MAPs (Markovian Arrival Processes) and considering the Drazin inverse, one can obtain explicit expressions for the y-intercept, together with some further insight regarding the BRAVO effect (Balancing Reduces Asymptotic Variance of Outputs). For M/G/1 queues our results are based on a classic transform of D.J. Daley. In this case we represent the y-intercept of the variance curve in terms of the first three moments of the service time distribution. A further performance measure that we are able to calculate for both models, is the asymptotic covariance between the queue length and the number of arrivals or departures. In addition we shed light on a classic conjecture of Daley, dealing with characterization of stationary M/M/1 queues within the class of stationary M/G/1 queues, based on the variance curve.
We consider open multi-class queueing networks with general arrival processes, general processing... more We consider open multi-class queueing networks with general arrival processes, general processing time sequences and Bernoulli routing. The network is assumed to be operating under an arbitrary work-conserving scheduling policy that makes the system stable. We study the variability of flows within the network. Computable expressions for quantifying flow variability have previously been discussed in the literature. However, in this paper, we shed more light on such analysis to justify the use of these expressions in the asymptotic analysis of network flows. Towards that end, we find a simple diffusion limit for the inter-class flows and establish the relation to asymptotic (co-)variance rates.
We study critical GI/G/1 queues under finite second moment assumptions. We show that the busy per... more We study critical GI/G/1 queues under finite second moment assumptions. We show that the busy period distribution is regularly varying with index half. We also review previously known M/G/1/ and M/M/1 derivations, yielding exact asymptotics as well as a similar derivation for GI/M/1. The busy period asymptotics determine the growth rate of moments of the renewal process counting busy cycles. We further use this to demonstrate a BRAVO phenomenon (Balancing Reduces Asymptotic Variance of Outputs) for the work-output process (namely the busy-time). This yields new insight on the BRAVO effect. A second contribution of the paper is in settling previous conjectured results about GI/G/1 and GI/G/s BRAVO. Previously, infinite buffer BRAVO was generally only settled under fourth-moment assumptions together with an assumption about the tail of the busy-period. In the current paper we strengthen the previous results by reducing to assumptions to existence of 2 + ǫ moments.
We consider a push pull queueing system with two servers and two types of jobs which are processe... more We consider a push pull queueing system with two servers and two types of jobs which are processed by the two servers in opposite order, with stochastic generally distributed processing times. This push pull system was introduced by Kopzon and Weiss, who assumed exponential processing times. It is similar to the Kumar-Seidman Rybko-Stolyar (KSRS) multi-class queueing network, with the distinction that instead of random arrivals, there is an infinite supply of jobs of both types. Thus each server can either process jobs of one of the types, which it pulls from the other server, or jobs of the other type which it pushes out of the infinite supply towards the other server. Unlike the KSRS network, we can find policies under which our push pull network works at full utilization, with both servers busy at all times, and without being congested. We perform an asymptotic analysis of the push pull network under these policies to quantify its behavior: We show that under fluid scaling the fluid model of the network is stable. We adapt the proofs of Dai, to show that as a result the queues of jobs waiting for pull operation are positive Harris recurrent. Finally we obtain the diffusion scale behavior of the network, in which we show that the queues are zero under diffusion scaling, and calculate the Brownian approximation of the output processes of the two types of jobs. The approximation shows that the two output streams are highly negatively correlated.
We describe an experimental setup and a currently running experiment for evaluating how physical ... more We describe an experimental setup and a currently running experiment for evaluating how physical interactions over time and between individuals affect the spread of epidemics. Our experiment involves the voluntary use of the Safe Blues Android app by participants at The University of Auckland (UoA) City Campus in New Zealand. The app spreads multiple virtual safe virus strands via Bluetooth depending on the physical proximity of the subjects. The evolution of the virtual epidemics is recorded as they spread through the population. The data is presented as a real-time (and historical) dashboard. A simulation model is applied to calibrate strand parameters. Participants’ locations are not recorded, but participants are rewarded based on the duration of participation within a geofenced area, and aggregate participation numbers serve as part of the data. The 2021 experimental data is available as an open-source anonymized dataset, and once the experiment is complete, the remaining data ...
We consider open multi-class queueing networks with general arrival processes, general processing... more We consider open multi-class queueing networks with general arrival processes, general processing time sequences and Bernoulli routing. The network is assumed to be operating under an arbitrary work-conserving scheduling policy that makes the system stable. We study the variability of flows within the network. Computable expressions for quantifying flow variability have previously been discussed in the literature. However, in this paper, we shed more light on such analysis to justify the use of these expressions in the asymptotic analysis of network flows. Towards that end, we find a simple diffusion limit for the inter-class flows and establish the relation to asymptotic (co-)variance rates.
Folklore often treats the Markov Modulated Poisson Process as bursty because the variance divided... more Folklore often treats the Markov Modulated Poisson Process as bursty because the variance divided by the expectation of counts is greater than unity. When viewed through the lens of the inter-event process, this ideally corresponds to a squared coefficient of variation greater than unity. As this has not been proved to date, we provide a proof together with an associated stochastic order relation.
Probability in the Engineering and Informational Sciences
We consider a collection of statistically identical two-state continuous time Markov chains (chan... more We consider a collection of statistically identical two-state continuous time Markov chains (channels). A controller continuously selects a channel with the view of maximizing infinite horizon average reward. A switching cost is paid upon channel changes. We consider two cases: full observation (all channels observed simultaneously) and partial observation (only the current channel observed). We analyze the difference in performance between these cases for various policies. For the partial observation case with two channels or an infinite number of channels, we explicitly characterize an optimal threshold for two sensible policies which we name “call-gapping” and “cool-off.” Our results present a qualitative view on the interaction of the number of channels, the available information, and the switching costs.
We study critical GI/G/1 queues under finite second-moment assumptions. We show that the busy-per... more We study critical GI/G/1 queues under finite second-moment assumptions. We show that the busy-period distribution is regularly varying with index half. We also review previously known M/G/1/ and M/M/1 derivations, yielding exact asymptotics as well as a similar derivation for GI/M/1. The busy-period asymptotics determine the growth rate of moments of the renewal process counting busy cycles. We further use this to demonstrate a Balancing Reduces Asymptotic Variance of Outputs (BRAVO) phenomenon for the work-output process (namely the busy time). This yields new insight on the BRAVO effect. A second contribution of the paper is in settling previous conjectured results about GI/G/1 and GI/G/s BRAVO. Previously, infinite buffer BRAVO was generally only settled under fourth-moment assumptions together with an assumption about the tail of the busy period. In the current paper, we strengthen the previous results by reducing to assumptions to existence of $$2+\epsilon $$ 2 + ϵ moments.
Many real life queueing networks have finite buffers with overflows. To understand the behavior o... more Many real life queueing networks have finite buffers with overflows. To understand the behavior of such networks, we consider traffic equations that generalize the traffic equations of classic open queueing networks where some nodes are potentially overloaded. We present a novel, efficient algorithm for solving the equations for overflow networks together with a sufficient condition for existence and uniqueness of solutions. Our analysis also sharpens results of traffic equations for classic open queueing networks.
We consider two classes of irreducible Markovian arrival processes specified by the matrices C an... more We consider two classes of irreducible Markovian arrival processes specified by the matrices C and D: the Markov-modulated Poisson process (MMPP) and the Markov-switched Poisson process (MSPP). The former exhibits a diagonal matrix D while the latter exhibits a diagonal matrix C. For these two classes we consider the following four statements: (I) the counting process is overdispersed; (II) the hazard rate of the event-stationary interarrival time is nonincreasing; (III) the squared coefficient of variation of the event-stationary process is greater than or equal to one; (IV) there is a stochastic order showing that the time-stationary interarrival time dominates the event-stationary interarrival time. For general MSPPs and order two MMPPs, we show that (I)–(IV) hold. Then for general MMPPs, it is easy to establish (I), while (II) is shown to be false by a counter-example. For general simple point processes, (III) follows from (IV). For MMPPs, we conjecture that (IV) and thus (III) ...
We consider single-pass, lossless, queueing systems at steady-state subject to Poisson job arriva... more We consider single-pass, lossless, queueing systems at steady-state subject to Poisson job arrivals at an unknown rate. Service rates are allowed to depend on the number of jobs in the system, up to a fixed maximum, and power consumption is an increasing function of speed. The goal is to control the state dependent service rates such that both energy consumption and delay are kept low. We consider a linear combination of the mean job delay and energy consumption as the performance measure. We examine both the “architecture ” of the system, which we define as a specification of the number of speeds that the system can choose from, and the “design ” of the system, which we define as the actual speeds available. Previous work has illustrated, that when the arrival rate is precisely known, there is little benefit in introducing complex (multi-speed) architectures, yet in view of parameter uncertainty, allowing a variable number of speeds improves robustness. We quantify the tradeoffs of...
International Series in Operations Research & Management Science, 2017
Wireless devices are often able to communicate on several alternative channels; for example, cell... more Wireless devices are often able to communicate on several alternative channels; for example, cellular phones may use several frequency bands and are equipped with base-station communication capability together with WiFi and Bluetooth communication. Automatic decision support systems in such devices need to decide which channels to use at any given time so as to maximize the long-run average throughput. A good decision policy needs to take into account that, due to cost, energy, technical, or performance constraints, the state of a channel is only sensed when it is selected for transmission. Therefore, the greedy strategy of always exploiting those channels assumed to yield the currently highest transmission rate is not necessarily optimal with respect to long-run average throughput. Rather, it may be favourable to give some priority to the exploration of channels of uncertain quality. In this chapter we model such on-line control problems as a special type of Restless Multi-Armed Bandit (RMAB) problem in a partially observable Markov decision process framework. We refer to such models as Reward-Observing Restless Multi-Armed Bandit (RORMAB) problems. These types of optimal control problems were previously considered in the literature in the context of: (i) the Gilbert-Elliot (GE) channels (where channels are modelled as a two state Markov chain), and (ii) Gaussian autoregressive (AR) channels of order 1. A virtue of this chapter is that we unify the presentation of both types of models under the umbrella of our newly defined RORMAB. Further, since RORMAB is a special type of RMAB we also present
Fluid solutions of queueing networks over a finite time horizon often involve non-empty queues an... more Fluid solutions of queueing networks over a finite time horizon often involve non-empty queues and full utilization of some of the resources for substantial portions of the time horizon. This observation makes it appropriate to treat the queueing network for which the transient fluid solution is specified, as a network with infinite virtual queues (i.e. queues that never empty yet need to be served at specified nominal rates). Examples of such queueing networks have been previously analyzed under steady state by Weiss, Kopzon and Adan.
2015 5th Australian Control Conference (AUCC), 2015
We address the problem of stabilizing control for complex queueing systems where servers follow u... more We address the problem of stabilizing control for complex queueing systems where servers follow unobservable Markovian environments. The controller needs to assign servers to queues without full information about the servers' states. A control challenge is to devise a policy that matches servers to queues in a way that takes state estimates into account and updates these estimates in the best way possible. Maximally attainable stability regions are non-trivial. We present the model, the control problem, and some preliminary methods for analysis and control. We illustrate basic phenomena and then focus on the simplest possible model having a single queue, a fixed state server, and a two state server. For this case, we begin analysis of a partially observable Markov decision process (POMDP) hinting at some structural properties. We also show how to use a quasi-birth-death (QBD) process for analysis and control.
Stochastic models for performance analysis, optimization and control of queues hinge on a multitu... more Stochastic models for performance analysis, optimization and control of queues hinge on a multitude of alternatives for input point processes. In case of bursty traffic, one very popular model is the Markov Modulated Poisson Process (MMPP), however it is not the only option. Here, we introduce an alternative that we call Markovian transition counting process (MTCP). The latter is a point process counting the number of transitions of a finite continuous-time Markov chain. For a given MTCP one can establish an MMPP with the same first and second moments of counts. In this paper, we show the other direction by establishing a duality in terms of first and second moments of counts between MTCPs and a rich class of MMPPs which we refer to as slow MMPPs (modulation is slower than the events). Such a duality confirms the applicability of the MTCP as an alternative to the MMPP which is superior when it comes to moment matching and finding the important measures of the inter-event process. We...
We consider a simple discrete time controlled queueing system, where the controller has a choice ... more We consider a simple discrete time controlled queueing system, where the controller has a choice of which server to use at each time slot and server performance varies according to a Markov modulated random environment. We explore the role of information in the system *
We consider queueing output processes of some elementary queueing models such as the M/M/1/K queu... more We consider queueing output processes of some elementary queueing models such as the M/M/1/K queue and the M/G/1 queue. An important performance measure for these counting processes is their variance curve, indicating the variance of the number of served customers over a time interval. Recent work has revealed some non-trivial properties dealing with the asymptotic rate at which the variance curve grows. In this paper we add to the results by finding explicit expressions for the second order approximation of the variance curve, namely the y-intercept of the linear asymptote. For M/M/1/K queues our results are based on the Drazin inverse of the generator. It turns out that by viewing output processes as MAPs (Markovian Arrival Processes) and considering the Drazin inverse, one can obtain explicit expressions for the y-intercept, together with some further insight regarding the BRAVO effect (Balancing Reduces Asymptotic Variance of Outputs). For M/G/1 queues our results are based on a classic transform of D.J. Daley. In this case we represent the y-intercept of the variance curve in terms of the first three moments of the service time distribution. A further performance measure that we are able to calculate for both models, is the asymptotic covariance between the queue length and the number of arrivals or departures. In addition we shed light on a classic conjecture of Daley, dealing with characterization of stationary M/M/1 queues within the class of stationary M/G/1 queues, based on the variance curve.
We consider open multi-class queueing networks with general arrival processes, general processing... more We consider open multi-class queueing networks with general arrival processes, general processing time sequences and Bernoulli routing. The network is assumed to be operating under an arbitrary work-conserving scheduling policy that makes the system stable. We study the variability of flows within the network. Computable expressions for quantifying flow variability have previously been discussed in the literature. However, in this paper, we shed more light on such analysis to justify the use of these expressions in the asymptotic analysis of network flows. Towards that end, we find a simple diffusion limit for the inter-class flows and establish the relation to asymptotic (co-)variance rates.
We study critical GI/G/1 queues under finite second moment assumptions. We show that the busy per... more We study critical GI/G/1 queues under finite second moment assumptions. We show that the busy period distribution is regularly varying with index half. We also review previously known M/G/1/ and M/M/1 derivations, yielding exact asymptotics as well as a similar derivation for GI/M/1. The busy period asymptotics determine the growth rate of moments of the renewal process counting busy cycles. We further use this to demonstrate a BRAVO phenomenon (Balancing Reduces Asymptotic Variance of Outputs) for the work-output process (namely the busy-time). This yields new insight on the BRAVO effect. A second contribution of the paper is in settling previous conjectured results about GI/G/1 and GI/G/s BRAVO. Previously, infinite buffer BRAVO was generally only settled under fourth-moment assumptions together with an assumption about the tail of the busy-period. In the current paper we strengthen the previous results by reducing to assumptions to existence of 2 + ǫ moments.
We consider a push pull queueing system with two servers and two types of jobs which are processe... more We consider a push pull queueing system with two servers and two types of jobs which are processed by the two servers in opposite order, with stochastic generally distributed processing times. This push pull system was introduced by Kopzon and Weiss, who assumed exponential processing times. It is similar to the Kumar-Seidman Rybko-Stolyar (KSRS) multi-class queueing network, with the distinction that instead of random arrivals, there is an infinite supply of jobs of both types. Thus each server can either process jobs of one of the types, which it pulls from the other server, or jobs of the other type which it pushes out of the infinite supply towards the other server. Unlike the KSRS network, we can find policies under which our push pull network works at full utilization, with both servers busy at all times, and without being congested. We perform an asymptotic analysis of the push pull network under these policies to quantify its behavior: We show that under fluid scaling the fluid model of the network is stable. We adapt the proofs of Dai, to show that as a result the queues of jobs waiting for pull operation are positive Harris recurrent. Finally we obtain the diffusion scale behavior of the network, in which we show that the queues are zero under diffusion scaling, and calculate the Brownian approximation of the output processes of the two types of jobs. The approximation shows that the two output streams are highly negatively correlated.
We describe an experimental setup and a currently running experiment for evaluating how physical ... more We describe an experimental setup and a currently running experiment for evaluating how physical interactions over time and between individuals affect the spread of epidemics. Our experiment involves the voluntary use of the Safe Blues Android app by participants at The University of Auckland (UoA) City Campus in New Zealand. The app spreads multiple virtual safe virus strands via Bluetooth depending on the physical proximity of the subjects. The evolution of the virtual epidemics is recorded as they spread through the population. The data is presented as a real-time (and historical) dashboard. A simulation model is applied to calibrate strand parameters. Participants’ locations are not recorded, but participants are rewarded based on the duration of participation within a geofenced area, and aggregate participation numbers serve as part of the data. The 2021 experimental data is available as an open-source anonymized dataset, and once the experiment is complete, the remaining data ...
We consider open multi-class queueing networks with general arrival processes, general processing... more We consider open multi-class queueing networks with general arrival processes, general processing time sequences and Bernoulli routing. The network is assumed to be operating under an arbitrary work-conserving scheduling policy that makes the system stable. We study the variability of flows within the network. Computable expressions for quantifying flow variability have previously been discussed in the literature. However, in this paper, we shed more light on such analysis to justify the use of these expressions in the asymptotic analysis of network flows. Towards that end, we find a simple diffusion limit for the inter-class flows and establish the relation to asymptotic (co-)variance rates.
Folklore often treats the Markov Modulated Poisson Process as bursty because the variance divided... more Folklore often treats the Markov Modulated Poisson Process as bursty because the variance divided by the expectation of counts is greater than unity. When viewed through the lens of the inter-event process, this ideally corresponds to a squared coefficient of variation greater than unity. As this has not been proved to date, we provide a proof together with an associated stochastic order relation.
Probability in the Engineering and Informational Sciences
We consider a collection of statistically identical two-state continuous time Markov chains (chan... more We consider a collection of statistically identical two-state continuous time Markov chains (channels). A controller continuously selects a channel with the view of maximizing infinite horizon average reward. A switching cost is paid upon channel changes. We consider two cases: full observation (all channels observed simultaneously) and partial observation (only the current channel observed). We analyze the difference in performance between these cases for various policies. For the partial observation case with two channels or an infinite number of channels, we explicitly characterize an optimal threshold for two sensible policies which we name “call-gapping” and “cool-off.” Our results present a qualitative view on the interaction of the number of channels, the available information, and the switching costs.
We study critical GI/G/1 queues under finite second-moment assumptions. We show that the busy-per... more We study critical GI/G/1 queues under finite second-moment assumptions. We show that the busy-period distribution is regularly varying with index half. We also review previously known M/G/1/ and M/M/1 derivations, yielding exact asymptotics as well as a similar derivation for GI/M/1. The busy-period asymptotics determine the growth rate of moments of the renewal process counting busy cycles. We further use this to demonstrate a Balancing Reduces Asymptotic Variance of Outputs (BRAVO) phenomenon for the work-output process (namely the busy time). This yields new insight on the BRAVO effect. A second contribution of the paper is in settling previous conjectured results about GI/G/1 and GI/G/s BRAVO. Previously, infinite buffer BRAVO was generally only settled under fourth-moment assumptions together with an assumption about the tail of the busy period. In the current paper, we strengthen the previous results by reducing to assumptions to existence of $$2+\epsilon $$ 2 + ϵ moments.
Many real life queueing networks have finite buffers with overflows. To understand the behavior o... more Many real life queueing networks have finite buffers with overflows. To understand the behavior of such networks, we consider traffic equations that generalize the traffic equations of classic open queueing networks where some nodes are potentially overloaded. We present a novel, efficient algorithm for solving the equations for overflow networks together with a sufficient condition for existence and uniqueness of solutions. Our analysis also sharpens results of traffic equations for classic open queueing networks.
We consider two classes of irreducible Markovian arrival processes specified by the matrices C an... more We consider two classes of irreducible Markovian arrival processes specified by the matrices C and D: the Markov-modulated Poisson process (MMPP) and the Markov-switched Poisson process (MSPP). The former exhibits a diagonal matrix D while the latter exhibits a diagonal matrix C. For these two classes we consider the following four statements: (I) the counting process is overdispersed; (II) the hazard rate of the event-stationary interarrival time is nonincreasing; (III) the squared coefficient of variation of the event-stationary process is greater than or equal to one; (IV) there is a stochastic order showing that the time-stationary interarrival time dominates the event-stationary interarrival time. For general MSPPs and order two MMPPs, we show that (I)–(IV) hold. Then for general MMPPs, it is easy to establish (I), while (II) is shown to be false by a counter-example. For general simple point processes, (III) follows from (IV). For MMPPs, we conjecture that (IV) and thus (III) ...
We consider single-pass, lossless, queueing systems at steady-state subject to Poisson job arriva... more We consider single-pass, lossless, queueing systems at steady-state subject to Poisson job arrivals at an unknown rate. Service rates are allowed to depend on the number of jobs in the system, up to a fixed maximum, and power consumption is an increasing function of speed. The goal is to control the state dependent service rates such that both energy consumption and delay are kept low. We consider a linear combination of the mean job delay and energy consumption as the performance measure. We examine both the “architecture ” of the system, which we define as a specification of the number of speeds that the system can choose from, and the “design ” of the system, which we define as the actual speeds available. Previous work has illustrated, that when the arrival rate is precisely known, there is little benefit in introducing complex (multi-speed) architectures, yet in view of parameter uncertainty, allowing a variable number of speeds improves robustness. We quantify the tradeoffs of...
International Series in Operations Research & Management Science, 2017
Wireless devices are often able to communicate on several alternative channels; for example, cell... more Wireless devices are often able to communicate on several alternative channels; for example, cellular phones may use several frequency bands and are equipped with base-station communication capability together with WiFi and Bluetooth communication. Automatic decision support systems in such devices need to decide which channels to use at any given time so as to maximize the long-run average throughput. A good decision policy needs to take into account that, due to cost, energy, technical, or performance constraints, the state of a channel is only sensed when it is selected for transmission. Therefore, the greedy strategy of always exploiting those channels assumed to yield the currently highest transmission rate is not necessarily optimal with respect to long-run average throughput. Rather, it may be favourable to give some priority to the exploration of channels of uncertain quality. In this chapter we model such on-line control problems as a special type of Restless Multi-Armed Bandit (RMAB) problem in a partially observable Markov decision process framework. We refer to such models as Reward-Observing Restless Multi-Armed Bandit (RORMAB) problems. These types of optimal control problems were previously considered in the literature in the context of: (i) the Gilbert-Elliot (GE) channels (where channels are modelled as a two state Markov chain), and (ii) Gaussian autoregressive (AR) channels of order 1. A virtue of this chapter is that we unify the presentation of both types of models under the umbrella of our newly defined RORMAB. Further, since RORMAB is a special type of RMAB we also present
Fluid solutions of queueing networks over a finite time horizon often involve non-empty queues an... more Fluid solutions of queueing networks over a finite time horizon often involve non-empty queues and full utilization of some of the resources for substantial portions of the time horizon. This observation makes it appropriate to treat the queueing network for which the transient fluid solution is specified, as a network with infinite virtual queues (i.e. queues that never empty yet need to be served at specified nominal rates). Examples of such queueing networks have been previously analyzed under steady state by Weiss, Kopzon and Adan.
2015 5th Australian Control Conference (AUCC), 2015
We address the problem of stabilizing control for complex queueing systems where servers follow u... more We address the problem of stabilizing control for complex queueing systems where servers follow unobservable Markovian environments. The controller needs to assign servers to queues without full information about the servers' states. A control challenge is to devise a policy that matches servers to queues in a way that takes state estimates into account and updates these estimates in the best way possible. Maximally attainable stability regions are non-trivial. We present the model, the control problem, and some preliminary methods for analysis and control. We illustrate basic phenomena and then focus on the simplest possible model having a single queue, a fixed state server, and a two state server. For this case, we begin analysis of a partially observable Markov decision process (POMDP) hinting at some structural properties. We also show how to use a quasi-birth-death (QBD) process for analysis and control.
Stochastic models for performance analysis, optimization and control of queues hinge on a multitu... more Stochastic models for performance analysis, optimization and control of queues hinge on a multitude of alternatives for input point processes. In case of bursty traffic, one very popular model is the Markov Modulated Poisson Process (MMPP), however it is not the only option. Here, we introduce an alternative that we call Markovian transition counting process (MTCP). The latter is a point process counting the number of transitions of a finite continuous-time Markov chain. For a given MTCP one can establish an MMPP with the same first and second moments of counts. In this paper, we show the other direction by establishing a duality in terms of first and second moments of counts between MTCPs and a rich class of MMPPs which we refer to as slow MMPPs (modulation is slower than the events). Such a duality confirms the applicability of the MTCP as an alternative to the MMPP which is superior when it comes to moment matching and finding the important measures of the inter-event process. We...
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Papers by Yoni Nazarathy