Discrete-time Markovian arrival processes to model multi-state complex systems with loss of units and an indeterminate variable number of repairpersons

This paper presents an algorithmic procedure to calculate the delay distribution of a type k customer in a first-come-first-served (FCFS) discrete-time queueing system with multiple types of customers, where each type has different service requirements (the MMAP[K]/PH[K]/1 queue). First, we develop a procedure, using matrix analytical methods, to handle arrival processes that do not allow batch arrivals to occur. Next, we show that this technique can be generalized to arrival processes that do allow batch arrivals to occur. We end the paper by presenting some numerical examples.

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Performance analysis of statistical multiplexing of heterogeneous discrete-time Markovian arrival processes in an ATM network

Computer Communications ◽

10.1016/s0140-3664(97)00098-4 ◽

1997 ◽

Vol 20 (11) ◽

pp. 970-978 ◽

Cited By ~ 9

Author(s):

Koohong Kang ◽

Cheeha Kim

Keyword(s):

Performance Analysis ◽

Discrete Time ◽

Statistical Multiplexing ◽

Atm Network ◽

Arrival Processes

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Part IV: 11 Comparison of queues with discrete-time arrival processes

Lecture Notes in Mathematics - Discrete-Event Control of Stochastic Networks: Multimodularity and Regularity ◽

10.1007/978-3-540-39705-2_12 ◽

2003 ◽

pp. 229-241

Author(s):

Eitan Altman ◽

Bruno Gaujal ◽

Arie Hordijk

Keyword(s):

Discrete Time ◽

Arrival Processes

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Exact queueing analysis of discrete time tandems with arbitrary arrival processes

2004 IEEE International Conference on Communications (IEEE Cat. No.04CH37577) ◽

10.1109/icc.2004.1312912 ◽

2004 ◽

Cited By ~ 5

Author(s):

M.J. Neely

Keyword(s):

Discrete Time ◽

Queueing Analysis ◽

Arrival Processes

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COMPARISON OF QUEUES WITH DIFFERENT DISCRETE-TIME ARRIVAL PROCESSES

Probability in the Engineering and Informational Sciences ◽

10.1017/s0269964801151016 ◽

2001 ◽

Vol 15 (1) ◽

pp. 1-14 ◽

Cited By ~ 3

Author(s):

Arie Hordijk

Keyword(s):

Lower Bound ◽

Discrete Time ◽

Sojourn Time ◽

Arrival Process ◽

Convex Ordering ◽

Arrival Processes ◽

Markov Arrival ◽

Markov Arrival Process ◽

Special Case ◽

Stochastic Event

Traveling times in a FIFO-stochastic event graph are compared in increasing convex ordering for different arrival processes. As a special case, a stochastic lower bound is obtained for the sojourn time in a tandem network of FIFO queues with a Markov arrival process. A counterexample shows that the extended Ross conjecture is not true for discrete-time arrival processes.

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Memoryless Systems Generate the Class of all Discrete Systems

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/2019/6803526 ◽

2019 ◽

Vol 2019 ◽

pp. 1-10

Author(s):

Erwan Beurier ◽

Dominique Pastor ◽

David I. Spivak

Keyword(s):

Complex Systems ◽

Discrete Time ◽

Internal State ◽

Biological Systems ◽

Discrete Systems ◽

Building Blocks ◽

Simple Cells ◽

Simple Machines ◽

Fixed Set ◽

The Brain

Automata are machines, which receive inputs, accordingly update their internal state, and produce output, and are a common abstraction for the basic building blocks used in engineering and science to describe and design complex systems. These arbitrarily simple machines can be wired together—so that the output of one is passed to another as its input—to form more complex machines. Indeed, both modern computers and biological systems can be described in this way, as assemblies of transistors or assemblies of simple cells. The complexity is in the network, i.e., the connection patterns between simple machines. The main result of this paper is to show that the range of simplicity for parts as compared to the complexity for wholes is in some sense complete: the most complex automaton can be obtained by wiring together direct-output memoryless components. The model we use—discrete-time automata sending each other messages from a fixed set of possibilities—is certainly more appropriate for computer systems than for biological systems. However, the result leads one to wonder what might be the simplest sort of machines, broadly construed, that can be assembled to produce the behaviour found in biological systems, including the brain.

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