21 research outputs found
Π₯Π°ΡΠ°ΠΊΡΠ΅ΡΠΈΡΡΠΈΠΊΠΈ ΠΏΡΠΎΠΈΠ·Π²ΠΎΠ΄ΠΈΡΠ΅Π»ΡΠ½ΠΎΡΡΠΈ ΡΠΈΡΡΠ΅ΠΌΡ ΠΌΠ°ΡΡΠΎΠ²ΠΎΠ³ΠΎ ΠΎΠ±ΡΠ»ΡΠΆΠΈΠ²Π°Π½ΠΈΡ Ρ ΡΠ°ΡΡΠ΅ΠΏΠ»Π΅Π½ΠΈΠ΅ΠΌ Π·Π°ΠΏΡΠΎΡΠΎΠ²
Objectives. The problem of investigating a fork-join queuing system is considered. It is required to build the process of the system functioning, to find the condition for the existence of a stationary distribution, and propose algorithms for calculating the stationary distribution and the main stationary performance characteristics. The special interest of the study is to obtain the lower and upper bounds of the mean sojourn time of a customer in the system.Methods. Methods of probability theory, queuing theory and matrix theory are used.Results. The functioning of the system is described in terms of a multidimensional Markov chain. A constructive condition for the existence of a stationary distribution is found, and algorithms for calculating the stationary distribution and stationary performance characteristics of the system are proposed. Analytical expressions are obtained for the lower and upper bounds of the mean sojourn time of customers in the system.Conclusion. The functioning of the fork-join queuing system with a stationary Poisson flow has been studied. Each of the arriving customers forks into two tasks that go to two subsystems, each of which consists of a server and a buffer. We assume that the buffer to one of the servers is unlimited, and to the second server has a finite volume. Service times have, generally speaking, different phase distributions (PH-Phase type distributions). For this system, a condition for the existence of a stationary distribution is obtained, algorithms for calculating the stationary distribution and a number of stationary performance measures of the system are proposed. Analytical expressions for the lower and upper bounds of the mean sojourn time of a customer in the system from the moment it enters the system to the moment of synchronization, which is a critical performance indicator of the fork-join queues, are obtained. The results of the study can be used for modeling various computer and communication systems, in particular, systems that perform parallel computing, customer processing in distributed databases, and parallel disk access.Π¦Π΅Π»ΠΈ. Π Π°ΡΡΠΌΠ°ΡΡΠΈΠ²Π°Π΅ΡΡΡ Π·Π°Π΄Π°ΡΠ° ΠΏΠΎΡΡΡΠΎΠ΅Π½ΠΈΡ ΠΈ ΠΈΡΡΠ»Π΅Π΄ΠΎΠ²Π°Π½ΠΈΡ ΠΌΠ°ΡΠ΅ΠΌΠ°ΡΠΈΡΠ΅ΡΠΊΠΎΠΉ ΠΌΠΎΠ΄Π΅Π»ΠΈ ΡΡΠΎΡ
Π°ΡΡΠΈΡΠ΅ΡΠΊΠΎΠΉ ΡΠΈΡΡΠ΅ΠΌΡ Ρ ΡΠ°ΡΡΠ΅ΠΏΠ»Π΅Π½ΠΈΠ΅ΠΌ ΠΈ ΡΠ±ΠΎΡΠΊΠΎΠΉ Π·Π°ΠΏΡΠΎΡΠΎΠ². Π’ΡΠ΅Π±ΡΠ΅ΡΡΡ ΠΏΠΎΡΡΡΠΎΠΈΡΡ ΠΏΡΠΎΡΠ΅ΡΡ ΡΡΠ½ΠΊΡΠΈΠΎΠ½ΠΈΡΠΎΠ²Π°Π½ΠΈΡ ΡΠΈΡΡΠ΅ΠΌΡ, Π½Π°ΠΉΡΠΈ ΡΡΠ»ΠΎΠ²ΠΈΠ΅ ΡΡΡΠ΅ΡΡΠ²ΠΎΠ²Π°Π½ΠΈΡ ΡΡΠ°ΡΠΈΠΎΠ½Π°ΡΠ½ΠΎΠ³ΠΎ ΡΠ°ΡΠΏΡΠ΅Π΄Π΅Π»Π΅Π½ΠΈΡ, ΠΏΡΠ΅Π΄Π»ΠΎΠΆΠΈΡΡ Π°Π»Π³ΠΎΡΠΈΡΠΌΡ Π΅Π³ΠΎ Π²ΡΡΠΈΡΠ»Π΅Π½ΠΈΡ ΠΈ ΠΎΡΠ½ΠΎΠ²Π½ΡΡ
ΡΡΠ°ΡΠΈΠΎΠ½Π°ΡΠ½ΡΡ
Ρ
Π°ΡΠ°ΠΊΡΠ΅ΡΠΈΡΡΠΈΠΊ ΠΏΡΠΎΠΈΠ·Π²ΠΎΠ΄ΠΈΡΠ΅Π»ΡΠ½ΠΎΡΡΠΈ ΡΠΈΡΡΠ΅ΠΌΡ. ΠΡΠΎΠ±ΡΠΉ ΠΈΠ½ΡΠ΅ΡΠ΅Ρ Π²ΡΠ·ΡΠ²Π°Π΅Ρ Π·Π°Π΄Π°ΡΠ° ΠΏΠΎΠ»ΡΡΠ΅Π½ΠΈΡ Π½ΠΈΠΆΠ½Π΅ΠΉ ΠΈ Π²Π΅ΡΡ
Π½Π΅ΠΉ Π³ΡΠ°Π½ΠΈΡ ΠΌΠ°ΡΠ΅ΠΌΠ°ΡΠΈΡΠ΅ΡΠΊΠΎΠ³ΠΎ ΠΎΠΆΠΈΠ΄Π°Π½ΠΈΡ Π²ΡΠ΅ΠΌΠ΅Π½ΠΈ ΠΏΡΠ΅Π±ΡΠ²Π°Π½ΠΈΡ Π·Π°ΠΏΡΠΎΡΠ° Π² ΡΠΈΡΡΠ΅ΠΌΠ΅.ΠΠ΅ΡΠΎΠ΄Ρ. ΠΡΠΏΠΎΠ»ΡΠ·ΡΡΡΡΡ ΠΌΠ΅ΡΠΎΠ΄Ρ ΡΠ΅ΠΎΡΠΈΠΈ Π²Π΅ΡΠΎΡΡΠ½ΠΎΡΡΠ΅ΠΉ, ΡΠ΅ΠΎΡΠΈΠΈ ΠΌΠ°ΡΡΠΎΠ²ΠΎΠ³ΠΎ ΠΎΠ±ΡΠ»ΡΠΆΠΈΠ²Π°Π½ΠΈΡ ΠΈ ΡΠ΅ΠΎΡΠΈΠΈ ΠΌΠ°ΡΡΠΈΡ.Π Π΅Π·ΡΠ»ΡΡΠ°ΡΡ. Π€ΡΠ½ΠΊΡΠΈΠΎΠ½ΠΈΡΠΎΠ²Π°Π½ΠΈΠ΅ ΡΠΈΡΡΠ΅ΠΌΡ ΠΎΠΏΠΈΡΠ°Π½ΠΎ Π² ΡΠ΅ΡΠΌΠΈΠ½Π°Ρ
ΠΌΠ½ΠΎΠ³ΠΎΠΌΠ΅ΡΠ½ΠΎΠΉ ΡΠ΅ΠΏΠΈ ΠΠ°ΡΠΊΠΎΠ²Π°. ΠΠ°ΠΉΠ΄Π΅Π½ΠΎ ΠΊΠΎΠ½ΡΡΡΡΠΊΡΠΈΠ²Π½ΠΎΠ΅ ΡΡΠ»ΠΎΠ²ΠΈΠ΅ ΡΡΡΠ΅ΡΡΠ²ΠΎΠ²Π°Π½ΠΈΡ ΡΡΠ°ΡΠΈΠΎΠ½Π°ΡΠ½ΠΎΠ³ΠΎ ΡΠ°ΡΠΏΡΠ΅Π΄Π΅Π»Π΅Π½ΠΈΡ, ΠΏΡΠ΅Π΄Π»ΠΎΠΆΠ΅Π½Ρ Π°Π»Π³ΠΎΡΠΈΡΠΌΡ Π΅Π³ΠΎ Π²ΡΡΠΈΡΠ»Π΅Π½ΠΈΡ ΠΈ ΡΡΠ°ΡΠΈΠΎΠ½Π°ΡΠ½ΡΡ
Ρ
Π°ΡΠ°ΠΊΡΠ΅ΡΠΈΡΡΠΈΠΊ ΠΏΡΠΎΠΈΠ·Π²ΠΎΠ΄ΠΈΡΠ΅Π»ΡΠ½ΠΎΡΡΠΈ ΡΠΈΡΡΠ΅ΠΌΡ. ΠΠΎΠ»ΡΡΠ΅Π½Ρ Π°Π½Π°Π»ΠΈΡΠΈΡΠ΅ΡΠΊΠΈΠ΅ Π²ΡΡΠ°ΠΆΠ΅Π½ΠΈΡ Π΄Π»Ρ Π½ΠΈΠΆΠ½Π΅ΠΉ ΠΈ Π²Π΅ΡΡ
Π½Π΅ΠΉ Π³ΡΠ°Π½ΠΈΡ ΠΌΠ°ΡΠ΅ΠΌΠ°ΡΠΈΡΠ΅ΡΠΊΠΎΠ³ΠΎ ΠΎΠΆΠΈΠ΄Π°Π½ΠΈΡ Π²ΡΠ΅ΠΌΠ΅Π½ΠΈ ΠΏΡΠ΅Π±ΡΠ²Π°Π½ΠΈΡ Π·Π°ΠΏΡΠΎΡΠΎΠ² Π² ΡΠΈΡΡΠ΅ΠΌΠ΅.ΠΠ°ΠΊΠ»ΡΡΠ΅Π½ΠΈΠ΅. ΠΡΡΠ»Π΅Π΄ΠΎΠ²Π°Π½ ΡΡΠ°ΡΠΈΠΎΠ½Π°ΡΠ½ΡΠΉ ΡΠ΅ΠΆΠΈΠΌ ΡΡΠ½ΠΊΡΠΈΠΎΠ½ΠΈΡΠΎΠ²Π°Π½ΠΈΡ ΡΠΈΡΡΠ΅ΠΌΡ ΠΌΠ°ΡΡΠΎΠ²ΠΎΠ³ΠΎ ΠΎΠ±ΡΠ»ΡΠΆΠΈΠ²Π°Π½ΠΈΡ Ρ ΡΠ°ΡΡΠ΅ΠΏΠ»Π΅Π½ΠΈΠ΅ΠΌ ΠΈ ΡΠ±ΠΎΡΠΊΠΎΠΉ Π·Π°ΠΏΡΠΎΡΠΎΠ², ΠΏΠΎΡΡΡΠΏΠ°ΡΡΠΈΡ
Π² ΡΠΈΡΡΠ΅ΠΌΡ Π² ΡΡΠ°ΡΠΈΠΎΠ½Π°ΡΠ½ΠΎΠΌ ΠΏΡΠ°ΡΡΠΎΠ½ΠΎΠ²ΡΠΊΠΎΠΌ ΠΏΠΎΡΠΎΠΊΠ΅. ΠΠ°ΠΆΠ΄ΡΠΉ ΠΈΠ· ΠΏΠΎΡΡΡΠΏΠ°ΡΡΠΈΡ
Π·Π°ΠΏΡΠΎΡΠΎΠ² ΡΠ°ΡΡΠ΅ΠΏΠ»ΡΠ΅ΡΡΡ Π½Π° Π΄Π²Π° Π·Π°Π΄Π°Π½ΠΈΡ, ΠΊΠΎΡΠΎΡΡΠ΅ ΠΈΠ΄ΡΡ Π² Π΄Π²Π΅ ΠΏΠΎΠ΄ΡΠΈΡΡΠ΅ΠΌΡ, ΡΠΎΡΡΠΎΡΡΠΈΠ΅ ΠΈΠ· ΠΎΠ±ΡΠ»ΡΠΆΠΈΠ²Π°ΡΡΠ΅Π³ΠΎ ΠΏΡΠΈΠ±ΠΎΡΠ° ΠΈ Π±ΡΡΠ΅ΡΠ°. ΠΡΠ΅ΠΌΠ΅Π½Π° ΠΎΠ±ΡΠ»ΡΠΆΠΈΠ²Π°Π½ΠΈΡ Π·Π°Π΄Π°Π½ΠΈΠΉ ΠΈΠΌΠ΅ΡΡ ΡΠ°Π·Π½ΡΠ΅ ΡΠ°Π·ΠΎΠ²ΡΠ΅ ΡΠ°ΡΠΏΡΠ΅Π΄Π΅Π»Π΅Π½ΠΈΡ (PH-Phase type distributions). ΠΠ»Ρ Π΄Π°Π½Π½ΠΎΠΉ ΡΠΈΡΡΠ΅ΠΌΡ Π½Π°ΠΉΠ΄Π΅Π½ΠΎ ΡΡΠ»ΠΎΠ²ΠΈΠ΅ ΡΡΡΠ΅ΡΡΠ²ΠΎΠ²Π°Π½ΠΈΡ ΡΡΠ°ΡΠΈΠΎΠ½Π°ΡΠ½ΠΎΠ³ΠΎ ΡΠ°ΡΠΏΡΠ΅Π΄Π΅Π»Π΅Π½ΠΈΡ, ΠΏΡΠ΅Π΄Π»ΠΎΠΆΠ΅Π½Ρ Π°Π»Π³ΠΎΡΠΈΡΠΌΡ Π²ΡΡΠΈΡΠ»Π΅Π½ΠΈΡ ΡΡΠ°ΡΠΈΠΎΠ½Π°ΡΠ½ΠΎΠ³ΠΎ ΡΠ°ΡΠΏΡΠ΅Π΄Π΅Π»Π΅Π½ΠΈΡ ΠΈ ΡΡΠ΄Π° ΡΡΠ°ΡΠΈΠΎΠ½Π°ΡΠ½ΡΡ
Ρ
Π°ΡΠ°ΠΊΡΠ΅ΡΠΈΡΡΠΈΠΊ ΠΏΡΠΎΠΈΠ·Π²ΠΎΠ΄ΠΈΡΠ΅Π»ΡΠ½ΠΎΡΡΠΈ ΡΠΈΡΡΠ΅ΠΌΡ. ΠΠΎΠ»ΡΡΠ΅Π½Ρ Π°Π½Π°Π»ΠΈΡΠΈΡΠ΅ΡΠΊΠΈΠ΅ Π²ΡΡΠ°ΠΆΠ΅Π½ΠΈΡ Π΄Π»Ρ Π½ΠΈΠΆΠ½Π΅ΠΉ ΠΈ Π²Π΅ΡΡ
Π½Π΅ΠΉ Π³ΡΠ°Π½ΠΈΡ ΠΌΠ°ΡΠ΅ΠΌΠ°ΡΠΈΡΠ΅ΡΠΊΠΎΠ³ΠΎ ΠΎΠΆΠΈΠ΄Π°Π½ΠΈΡ Π²ΡΠ΅ΠΌΠ΅Π½ΠΈ ΠΏΡΠ΅Π±ΡΠ²Π°Π½ΠΈΡ Π·Π°ΠΏΡΠΎΡΠ° Π² ΡΠΈΡΡΠ΅ΠΌΠ΅ ΠΎΡ ΠΌΠΎΠΌΠ΅Π½ΡΠ° Π΅Π³ΠΎ ΠΏΠΎΡΡΡΠΏΠ»Π΅Π½ΠΈΡ Π² ΡΠΈΡΡΠ΅ΠΌΡ Π΄ΠΎ ΠΌΠΎΠΌΠ΅Π½ΡΠ° ΡΠΈΠ½Ρ
ΡΠΎΠ½ΠΈΠ·Π°ΡΠΈΠΈ, ΠΊΠΎΡΠΎΡΠΎΠ΅ ΡΠ²Π»ΡΠ΅ΡΡΡ ΠΊΡΠΈΡΠΈΡΠ΅ΡΠΊΠΈΠΌ ΠΏΠΎΠΊΠ°Π·Π°ΡΠ΅Π»Π΅ΠΌ ΠΏΡΠΎΠΈΠ·Π²ΠΎΠ΄ΠΈΡΠ΅Π»ΡΠ½ΠΎΡΡΠΈ ΡΠΈΡΡΠ΅ΠΌΡ Ρ ΡΠ°ΡΡΠ΅ΠΏΠ»Π΅Π½ΠΈΠ΅ΠΌ ΠΈ ΡΠ±ΠΎΡΠΊΠΎΠΉ Π·Π°ΠΏΡΠΎΡΠΎΠ²
LIPIcs, Volume 261, ICALP 2023, Complete Volume
LIPIcs, Volume 261, ICALP 2023, Complete Volum
LIPIcs, Volume 248, ISAAC 2022, Complete Volume
LIPIcs, Volume 248, ISAAC 2022, Complete Volum