22 research outputs found
Π‘ΠΠ’Π ΠΠΠ‘Π‘ΠΠΠΠΠ ΠΠΠ‘ΠΠ£ΠΠΠΠΠΠΠ― ΠΠ ΠΠΠΠΠΠΠ¬ΠΠΠ Π’ΠΠΠΠΠΠΠΠ Π‘ ΠΠΠΠΠΠΠΠ Π Π‘ΠΠΠ―ΠΠΠΠ Π’Π ΠΠΠΠΠΠΠΠ: Π‘ΠΠ£Π§ΠΠ ΠΠΠ‘ΠΠΠΠΠ§ΠΠΠΠ ΠΠΠΠ ΠΠ«Π₯ Π‘ΠΠ‘Π’ΠΠ ΠΠΠ‘ΠΠ£ΠΠΠΠΠΠΠ―
Π ΡΠ°Π±ΠΎΡΠ΅ ΡΠ°ΡΡΠΌΠ°ΡΡΠΈΠ²Π°Π΅ΡΡΡ ΠΊΠ»Π°ΡΡ ΠΎΡΠΊΡΡΡΡΡ
ΡΠ΅ΡΠ΅ΠΉ ΠΌΠ°ΡΡΠΎΠ²ΠΎΠ³ΠΎ ΠΎΠ±ΡΠ»ΡΠΆΠΈ-
Π²Π°Π½ΠΈΡ ΠΏΡΠΎΠΈΠ·Π²ΠΎΠ»ΡΠ½ΠΎΠΉ ΡΠΎΠΏΠΎΠ»ΠΎΠ³ΠΈΠΈ, ΠΊΠΎΡΠΎΡΡΠΉ ΡΠ²Π»ΡΠ΅ΡΡΡ ΡΠ°Π·Π²ΠΈΡΠΈΠ΅ΠΌ ΠΊΠ»Π°ΡΡΠΈΡΠ΅-
ΡΠΊΠΈΡ
fork-join ΡΠ΅ΡΠ΅ΠΉ ΠΌΠ°ΡΡΠΎΠ²ΠΎΠ³ΠΎ ΠΎΠ±ΡΠ»ΡΠΆΠΈΠ²Π°Π½ΠΈΡ, ΠΈΡΠΏΠΎΠ»ΡΠ·ΡΠ΅ΠΌΡΡ
Π² ΠΊΠ°ΡΠ΅ΡΡΠ²Π΅
ΠΌΠ°ΡΠ΅ΠΌΠ°ΡΠΈΡΠ΅ΡΠΊΠΈΡ
ΠΌΠΎΠ΄Π΅Π»Π΅ΠΉ ΡΡΠΎΡ
Π°ΡΡΠΈΡΠ΅ΡΠΊΠΈΡ
ΡΠΈΡΡΠ΅ΠΌ Ρ ΠΏΠ°ΡΠ°Π»Π»Π΅Π»ΡΠ½ΡΠΌ ΠΈ ΡΠ°Ρ-
ΠΏΡΠ΅Π΄Π΅Π»Π΅Π½Π½ΡΠΌ ΠΏΡΠΈΠ½ΡΠΈΠΏΠΎΠΌ ΡΡΠ½ΠΊΡΠΈΠΎΠ½ΠΈΡΠΎΠ²Π°Π½ΠΈΡ (GRID-ΡΠΈΡΡΠ΅ΠΌΡ, RAID-
ΠΌΠ°ΡΡΠΈΠ²Ρ, MapReduce ΠΈ Ρ. Π΄.)
Π‘ΠΈΡΡΠ΅ΠΌΡ ΠΎΠ±ΡΠ»ΡΠΆΠΈΠ²Π°Π½ΠΈΡ Π² ΡΠ°ΡΡΠΌΠ°ΡΡΠΈΠ²Π°Π΅ΠΌΠΎΠΉ ΡΠ΅ΡΠΈ ΠΏΠΎΠ΄Π΅Π»Π΅Π½Ρ Π½Π° ΡΡΠΈ ΡΠΈΠΏΠ°
Π² Π·Π°Π²ΠΈΡΠΈΠΌΠΎΡΡΠΈ ΠΎΡ ΠΈΡ
Π½Π°Π·Π½Π°ΡΠ΅Π½ΠΈΡ: Π±Π΅ΡΠΊΠΎΠ½Π΅ΡΠ½ΠΎΠΏΡΠΈΠ±ΠΎΡΠ½ΡΠ΅ Π±Π°Π·ΠΎΠ²ΡΠ΅ ΡΠΈΡΡΠ΅-
ΠΌΡ, Π΄ΠΈΠ²Π°ΠΉΠ΄Π΅ΡΡ, ΠΈΠ½ΡΠ΅Π³ΡΠ°ΡΠΎΡΡ. ΠΠ°Π»ΠΈΡΠΈΠ΅ Π±Π΅ΡΠΊΠΎΠ½Π΅ΡΠ½ΠΎΠ³ΠΎ ΡΠΈΡΠ»Π° ΠΎΠ±ΡΠ»ΡΠΆΠΈΠ²Π°-
ΡΡΠΈΡ
ΠΏΡΠΈΠ±ΠΎΡΠΎΠ² Π² Π±Π°Π·ΠΎΠ²ΡΡ
ΡΠΈΡΡΠ΅ΠΌΠ°Ρ
ΠΏΠΎΠ·Π²ΠΎΠ»ΡΠ΅Ρ ΡΡΡΠ΅ΡΡΠ²Π΅Π½Π½ΠΎ ΡΠΏΡΠΎΡΡΠΈΡΡ
Π°Π½Π°Π»ΠΈΠ· ΠΈ ΡΠ°ΡΡΠΌΠΎΡΡΠ΅ΡΡ ΡΠ΅ΡΠΈ ΠΎΠ±ΡΠ»ΡΠΆΠΈΠ²Π°Π½ΠΈΡ Ρ ΠΏΡΠΎΠΈΠ·Π²ΠΎΠ»ΡΠ½ΠΎΠΉ ΡΠΎΠΏΠΎΠ»ΠΎΠ³ΠΈΠ΅ΠΉ.
Π’ΡΠ΅Π±ΠΎΠ²Π°Π½ΠΈΠ΅, ΠΏΠΎΡΡΡΠΏΠ°ΡΡΠ΅Π΅ Π² Π΄ΠΈΠ²Π°ΠΉΠ΄Π΅Ρ, Π΄Π΅Π»ΠΈΡΡΡ Π½Π° Π½Π΅ΠΊΠΎΡΠΎΡΠΎΠ΅ ΡΠΈΡΠ»ΠΎ ΡΠ°ΡΡΠ΅ΠΉ β ΡΡΠ°Π³ΠΌΠ΅Π½ΡΠΎΠ². ΠΠΎΠ»ΡΡΠ΅Π½Π½ΡΠ΅ ΡΡΠ°Π³ΠΌΠ΅Π½ΡΡ ΠΎΠ±ΡΠ»ΡΠΆΠΈΠ²Π°ΡΡΡΡ Π½Π΅Π·Π°Π²ΠΈΡΠΈΠΌΠΎ Π΄ΡΡΠ³ ΠΎΡ Π΄ΡΡΠ³Π° Π² Π±Π°Π·ΠΎΠ²ΡΡ
ΡΠΈΡΡΠ΅ΠΌΠ°Ρ
ΡΠ΅ΡΠΈ, ΠΏΠ΅ΡΠ΅Ρ
ΠΎΠ΄ΡΡ ΠΏΠΎ ΡΠ΅ΡΠΈ. ΠΠ°ΠΆΠ΄ΡΠΉ
ΠΈΠ· ΡΡΠ°Π³ΠΌΠ΅Π½ΡΠΎΠ² ΠΌΠΎΠΆΠ΅Ρ ΡΠ½ΠΎΠ²Π° ΠΏΠΎΠ΄Π΅Π»ΠΈΡΡΡΡ Π½Π° ΡΡΠ°Π³ΠΌΠ΅Π½ΡΡ ΠΏΡΠΈ ΠΏΠΎΡΡΡΠΏΠ»Π΅Π½ΠΈΠΈ Π² Π΄ΠΈΠ²Π°ΠΉΠ΄Π΅Ρ. ΠΠ±ΡΠ΅Π΄ΠΈΠ½Π΅Π½ΠΈΠ΅ ΡΡΠ°Π³ΠΌΠ΅Π½ΡΠΎΠ² ΠΏΡΠΎΠΈΡΡ
ΠΎΠ΄ΠΈΡ Π² ΠΈΠ½ΡΠ΅Π³ΡΠ°ΡΠΎΡΠ°Ρ
. Π’Π°ΠΊ, ΠΏΠ΅ΡΠ΅Π΄ ΡΡ
ΠΎΠ΄ΠΎΠΌ ΠΈΠ· ΡΠ΅ΡΠΈ Π²ΡΠ΅ ΡΡΠ°Π³ΠΌΠ΅Π½ΡΡ ΠΎΠ±ΡΠ΅Π΄ΠΈΠ½ΡΡΡΡΡ Π² ΠΎΠ΄Π½ΠΎΠΌ ΠΈΠ· ΠΈΠ½ΡΠ΅Π³ΡΠ°ΡΠΎΡΠΎΠ² Π² ΠΈΡΡ
ΠΎΠ΄Π½ΠΎΠ΅ ΡΡΠ΅Π±ΠΎΠ²Π°Π½ΠΈΠ΅. ΠΠ»Π°Π²Π½ΡΠΌ ΡΠ΅Π·ΡΠ»ΡΡΠ°ΡΠΎΠΌ ΡΠ°Π±ΠΎΡΡ ΡΡΠ°Π» ΠΌΠ΅ΡΠΎΠ΄ ΠΏΠΎΠ»ΡΡΠ΅Π½ΠΈΡ ΡΠ°ΡΠΏΡΠ΅Π΄Π΅Π»Π΅Π½ΠΈΡ
Π΄Π»ΠΈΡΠ΅Π»ΡΠ½ΠΎΡΡΠΈ ΠΏΡΠ΅Π±ΡΠ²Π°Π½ΠΈΡ ΡΡΠ΅Π±ΠΎΠ²Π°Π½ΠΈΠΉ Π² ΠΈΠ·ΡΡΠ°Π΅ΠΌΠΎΠΉ ΡΠ΅ΡΠΈ ΠΌΠ°ΡΡΠΎΠ²ΠΎΠ³ΠΎ ΠΎΠ±ΡΠ»ΡΠΆΠΈΠ²Π°Π½ΠΈ
Approximations and Bounds for (n, k) Fork-Join Queues: A Linear Transformation Approach
Compared to basic fork-join queues, a job in (n, k) fork-join queues only
needs its k out of all n sub-tasks to be finished. Since (n, k) fork-join
queues are prevalent in popular distributed systems, erasure coding based cloud
storages, and modern network protocols like multipath routing, estimating the
sojourn time of such queues is thus critical for the performance measurement
and resource plan of computer clusters. However, the estimating keeps to be a
well-known open challenge for years, and only rough bounds for a limited range
of load factors have been given. In this paper, we developed a closed-form
linear transformation technique for jointly-identical random variables: An
order statistic can be represented by a linear combination of maxima. This
brand-new technique is then used to transform the sojourn time of non-purging
(n, k) fork-join queues into a linear combination of the sojourn times of basic
(k, k), (k+1, k+1), ..., (n, n) fork-join queues. Consequently, existing
approximations for basic fork-join queues can be bridged to the approximations
for non-purging (n, k) fork-join queues. The uncovered approximations are then
used to improve the upper bounds for purging (n, k) fork-join queues.
Simulation experiments show that this linear transformation approach is
practiced well for moderate n and relatively large k.Comment: 10 page
ΠΠ±Π·ΠΎΡ ΡΠΈΡΡΠ΅ΠΌ ΠΏΠ°ΡΠ°Π»Π»Π΅Π»ΡΠ½ΠΎΠΉ ΠΎΠ±ΡΠ°Π±ΠΎΡΠΊΠΈ Π·Π°ΡΠ²ΠΎΠΊ
This paper is the ο¬rst in a series of two articles devoted to the review of βfork-joinβ (inthe western classiο¬cation) queuing systems or systems with the splitting of incoming queries.This system is a natural model for many other real systems. The article describes the fork-joinqueueing model construction and main characteristics of this model. Special attention is paid tomethods of analysis of the response time of the system. Since the exact expression for the meanresponse time is known only for the case of two servers, the article gives a detailed descriptionof the approach to obtaining an accurate expression of this characteristic. For the case whenthe number of servers is more than two, approximations of the mean response time are obtainedby diο¬erent methods, which is explained by the complexity of the studies due to the existingdependence between the queues of subqueries due to common arrival moments. The paperpresents several methods of approximate analysis: various variants of empirical approximation,i.e. methods that reο¬ne the obtained characteristics by using the results of simulation modeling;interpolation methods using system load limit values in cases when the incoming ο¬ow and servicetime distributions are not exponential.ΠΠ°Π½Π½Π°Ρ ΡΠ°Π±ΠΎΡΠ° ΡΠ²Π»ΡΠ΅ΡΡΡ ΠΏΠ΅ΡΠ²ΠΎΠΉ Π² ΡΠ΅ΡΠΈΠΈ ΠΈΠ· Π΄Π²ΡΡ
ΡΡΠ°ΡΠ΅ΠΉ, ΠΏΠΎΡΠ²ΡΡΡΠ½Π½ΡΡ
ΠΎΠ±Π·ΠΎΡΡ ΡΠΈΡΡΠ΅ΠΌ ΠΌΠ°ΡΡΠΎΠ²ΠΎΠ³ΠΎ ΠΎΠ±ΡΠ»ΡΠΆΠΈΠ²Π°Π½ΠΈΡ Π²ΠΈΠ΄Π° Β«fork-joinΒ» (Π² Π·Π°ΠΏΠ°Π΄Π½ΠΎΠΉ ΠΊΠ»Π°ΡΡΠΈΡΠΈΠΊΠ°ΡΠΈΠΈ) ΠΈΠ»ΠΈ ΡΠΈΡΡΠ΅ΠΌΠ°ΠΌ Ρ ΡΠ°ΡΡΠ΅ΠΏΠ»Π΅Π½ΠΈΠ΅ΠΌ Π·Π°ΠΏΡΠΎΡΠΎΠ². Π£ΠΊΠ°Π·Π°Π½Π½Π°Ρ ΡΠΈΡΡΠ΅ΠΌΠ° ΡΠ²Π»ΡΠ΅ΡΡΡ Π΅ΡΡΠ΅ΡΡΠ²Π΅Π½Π½ΠΎΠΉ ΠΌΠΎΠ΄Π΅Π»ΡΡ Π΄Π»Ρ ΠΌΠ½ΠΎΠ³ΠΈΡ
Π΄ΡΡΠ³ΠΈΡ
ΡΠ΅Π°Π»ΡΠ½ΡΡ
ΡΠΈΡΡΠ΅ΠΌ. Π ΡΡΠ°ΡΡΠ΅ ΠΎΠΏΠΈΡΠ°Π½Ρ ΠΎΡΠΎΠ±Π΅Π½Π½ΠΎΡΡΠΈ ΠΏΠΎΡΡΡΠΎΠ΅Π½ΠΈΡ ΡΡΠΎΠΉ ΠΌΠΎΠ΄Π΅Π»ΠΈ ΠΈ ΡΠΎΠ΄ΡΡΠ²Π΅Π½Π½ΡΡ
Π΅ΠΉ ΡΠΈΡΡΠ΅ΠΌ, ΠΎΡΠ½ΠΎΠ²Π½ΡΠ΅ ΠΈΡ
Ρ
Π°ΡΠ°ΠΊΡΠ΅ΡΠΈΡΡΠΈΠΊΠΈ. ΠΡΠ΄Π΅Π»ΡΠ½ΠΎΠ΅ Π²Π½ΠΈΠΌΠ°Π½ΠΈΠ΅ ΡΠ΄Π΅Π»ΡΠ΅ΡΡΡ ΠΌΠ΅ΡΠΎΠ΄Π°ΠΌ Π°Π½Π°Π»ΠΈΠ·Π° Π²ΡΠ΅ΠΌΠ΅Π½ΠΈ ΠΎΡΠΊΠ»ΠΈΠΊΠ° ΡΠΈΡΡΠ΅ΠΌΡ. ΠΠΎΡΠΊΠΎΠ»ΡΠΊΡ ΡΠΎΡΠ½ΠΎΠ΅ Π²ΡΡΠ°ΠΆΠ΅Π½ΠΈΠ΅ Π΄Π»Ρ ΡΡΠ΅Π΄Π½Π΅Π³ΠΎ Π²ΡΠ΅ΠΌΠ΅Π½ΠΈ ΠΎΡΠΊΠ»ΠΈΠΊΠ° ΠΈΠ·Π²Π΅ΡΡΠ½ΠΎ ΡΠΎΠ»ΡΠΊΠΎ Π΄Π»Ρ ΡΠ»ΡΡΠ°Ρ Π΄Π²ΡΡ
ΠΏΡΠΈΠ±ΠΎΡΠΎΠ², Π² ΡΡΠ°ΡΡΠ΅ ΠΏΡΠΈΠ²Π΅Π΄Π΅Π½ΠΎ ΠΏΠΎΠ΄ΡΠΎΠ±Π½ΠΎΠ΅ ΠΎΠΏΠΈΡΠ°Π½ΠΈΠ΅ ΠΏΠΎΠ΄Ρ
ΠΎΠ΄Π° ΠΊ ΠΏΠΎΠ»ΡΡΠ΅Π½ΠΈΡ ΡΠΎΡΠ½ΠΎΠ³ΠΎ Π²ΡΡΠ°ΠΆΠ΅Π½ΠΈΡ ΡΡΠΎΠΉ Ρ
Π°ΡΠ°ΠΊΡΠ΅ΡΠΈΡΡΠΈΠΊΠΈ. ΠΠ»Ρ ΡΠ»ΡΡΠ°Ρ, ΠΊΠΎΠ³Π΄Π° ΡΠΈΡΠ»ΠΎ ΠΏΡΠΈΠ±ΠΎΡΠΎΠ² Π±ΠΎΠ»ΡΡΠ΅ Π΄Π²ΡΡ
, ΡΠ°Π·Π»ΠΈΡΠ½ΡΠΌΠΈ ΠΌΠ΅ΡΠΎΠ΄Π°ΠΌΠΈ ΠΏΠΎΠ»ΡΡΠ΅Π½Ρ Π°ΠΏΠΏΡΠΎΠΊΡΠΈΠΌΠ°ΡΠΈΠΈ ΡΡΠ΅Π΄Π½Π΅Π³ΠΎ Π²ΡΠ΅ΠΌΠ΅Π½ΠΈ ΠΎΡΠΊΠ»ΠΈΠΊΠ°,ΡΡΠΎ ΠΎΠ±ΡΡΡΠ½ΡΠ΅ΡΡΡ ΡΠ»ΠΎΠΆΠ½ΠΎΡΡΡΡ ΠΈΡΡΠ»Π΅Π΄ΠΎΠ²Π°Π½ΠΈΠΉ ΠΈΠ·-Π·Π° ΡΡΡΠ΅ΡΡΠ²ΡΡΡΠ΅ΠΉ Π·Π°Π²ΠΈΡΠΈΠΌΠΎΡΡΠΈ ΠΌΠ΅ΠΆΠ΄Ρ ΠΎΡΠ΅ΡΠ΅Π΄ΡΠΌΠΈ ΠΏΠΎΠ΄ Π·Π°ΠΏΡΠΎΡΠΎΠ² Π² ΡΠΈΠ»Ρ ΠΎΠ±ΡΠΈΡ
ΠΌΠΎΠΌΠ΅Π½ΡΠΎΠ² ΠΏΠΎΡΡΡΠΏΠ»Π΅Π½ΠΈΡ. Π ΡΠ°Π±ΠΎΡΠ΅ ΠΏΡΠ΅Π΄ΡΡΠ°Π²Π»Π΅Π½ΠΎ Π½Π΅ΡΠΊΠΎΠ»ΡΠΊΠΎ ΠΌΠ΅ΡΠΎΠ΄ΠΎΠ² ΠΏΡΠΈΠ±Π»ΠΈΠΆΠ΅Π½Π½ΠΎΠ³ΠΎ Π°Π½Π°Π»ΠΈΠ·Π°: ΡΠ°Π·Π»ΠΈΡΠ½ΡΠ΅ Π²Π°ΡΠΈΠ°Π½ΡΡ ΡΠΌΠΏΠΈΡΠΈΡΠ΅ΡΠΊΠΎΠΉ Π°ΠΏΠΏΡΠΎΠΊΡΠΈΠΌΠ°ΡΠΈΠΈ, Ρ.Π΅. ΠΌΠ΅ΡΠΎΠ΄Ρ, ΡΡΠΎΡΠ½ΡΡΡΠΈΠ΅ ΠΏΠΎΠ»ΡΡΠ΅Π½Π½ΡΠ΅ Ρ
Π°ΡΠ°ΠΊΡΠ΅ΡΠΈΡΡΠΈΠΊΠΈ Π±Π»Π°Π³ΠΎΠ΄Π°ΡΡ ΠΈΡΠΏΠΎΠ»ΡΠ·ΠΎΠ²Π°Π½ΠΈΡ ΡΠ΅Π·ΡΠ»ΡΡΠ°ΡΠΎΠ² ΠΈΠΌΠΈΡΠ°ΡΠΈΠΎΠ½Π½ΠΎΠ³ΠΎ ΠΌΠΎΠ΄Π΅Π»ΠΈΡΠΎΠ²Π°Π½ΠΈΡ; ΠΈΠ½ΡΠ΅ΡΠΏΠΎΠ»ΡΡΠΈΡ Ρ ΠΏΠΎΠΌΠΎΡΡΡ ΠΏΡΠ΅Π΄Π΅Π»ΡΠ½ΡΡ
Π·Π½Π°ΡΠ΅Π½ΠΈΠΉ Π·Π°Π³ΡΡΠ·ΠΊΠΈ ΡΠΈΡΡΠ΅ΠΌΡ Π² ΡΠ»ΡΡΠ°ΡΡ
Ρ ΠΎΡΠ»ΠΈΡΠ½ΡΠΌΠΈ ΠΎΡ ΡΠΊΡΠΏΠΎΠ½Π΅Π½ΡΠΈΠ°Π»ΡΠ½ΠΎΠ³ΠΎ ΡΠ°ΡΠΏΡΠ΅Π΄Π΅Π»Π΅Π½ΠΈΡΠΌΠΈ Π΄Π»Ρ Π²Ρ
ΠΎΠ΄ΡΡΠ΅Π³ΠΎ ΠΏΠΎΡΠΎΠΊΠ° ΠΈ Π²ΡΠ΅ΠΌΠ΅Π½ΠΈ ΠΎΠ±ΡΠ»ΡΠΆΠΈΠ²Π°Π½ΠΈΡ
ΠΠ±Π·ΠΎΡ ΡΠΈΡΡΠ΅ΠΌ ΠΏΠ°ΡΠ°Π»Π»Π΅Π»ΡΠ½ΠΎΠΉ ΠΎΠ±ΡΠ°Π±ΠΎΡΠΊΠΈ Π·Π°ΡΠ²ΠΎΠΊ. Π§Π°ΡΡΡ II
This paper is a continuation of the survey of the βfork-joinβ queuing systems (in the westernclassiο¬cation) or the systems with splitting of queries. Interest in such systems is explainedby a wide range of problems that can be solved with their help, since in fact it is a matter ofparallel processing of data and their applications. For example, this may concern the analysis ofdisk arrays, cloud computing, high-performance services and even the process of picking ordersin a warehouse. In the ο¬rst part of the survey, the main features of the described model (andrelated systems) and its construction were introduced. Also the detailed description of theapproach to obtaining an accurate expression of the average response time in the case of twodevices was presented as well as several methods of approximate analysis of this characteristic(the case when the number of devices is more than two). This part of the survey is devotedto the description of other existing methods for approximating the average response time. Inparticular, the approaches of the approximate analysis of the response time are as follows: thematrix-geometric method, the analysis with the help of order statistics for various types ofdistribution of the service time of subqueries.ΠΠ°Π½Π½Π°Ρ ΡΠ°Π±ΠΎΡΠ° ΡΠ²Π»ΡΠ΅ΡΡΡ ΠΏΡΠΎΠ΄ΠΎΠ»ΠΆΠ΅Π½ΠΈΠ΅ΠΌ ΠΎΠ±Π·ΠΎΡΠ° ΠΌΠ΅ΡΠΎΠ΄ΠΎΠ² ΠΈΡΡΠ»Π΅Π΄ΠΎΠ²Π°Π½ΠΈΡ ΡΠΈΡΡΠ΅ΠΌΡ ΠΌΠ°ΡΡΠΎΠ²ΠΎΠ³ΠΎΠΎΠ±ΡΠ»ΡΠΆΠΈΠ²Π°Π½ΠΈΡ Π²ΠΈΠ΄Π° Β«fork-joinΒ» (Π² Π·Π°ΠΏΠ°Π΄Π½ΠΎΠΉ ΠΊΠ»Π°ΡΡΠΈΡΠΈΠΊΠ°ΡΠΈΠΈ) ΠΈΠ»ΠΈ ΡΠΈΡΡΠ΅ΠΌΡ Ρ ΡΠ°ΡΡΠ΅ΠΏΠ»Π΅Π½ΠΈΠ΅ΠΌΠ·Π°ΠΏΡΠΎΡΠΎΠ². ΠΠ½ΡΠ΅ΡΠ΅Ρ ΠΊ ΡΠ°ΡΡΠΌΠ°ΡΡΠΈΠ²Π°Π΅ΠΌΠΎΠΉ ΡΠΈΡΡΠ΅ΠΌΠ΅ ΠΎΠ±ΡΡΡΠ½ΡΠ΅ΡΡΡ ΡΠΈΡΠΎΠΊΠΈΠΌ ΡΠΏΠ΅ΠΊΡΡΠΎΠΌ Π·Π°Π΄Π°Ρ, ΠΊΠΎΡΠΎΡΡΠ΅ ΠΌΠΎΠ³ΡΡ Π±ΡΡΡ ΡΠ΅ΡΠ΅Π½Ρ Ρ Π΅Ρ ΠΏΠΎΠΌΠΎΡΡΡ, ΠΏΠΎΡΠΊΠΎΠ»ΡΠΊΡ ΡΠ°ΠΊΡΠΈΡΠ΅ΡΠΊΠΈ ΡΠ΅ΡΡ ΠΈΠ΄ΡΡ ΠΎ ΠΏΠ°ΡΠ°Π»Π»Π΅Π»ΡΠ½ΠΎΠΉΠΎΠ±ΡΠ°Π±ΠΎΡΠΊΠ΅ Π΄Π°Π½Π½ΡΡ
ΠΈ ΠΈΡ
ΠΏΡΠΈΠ»ΠΎΠΆΠ΅Π½ΠΈΡΡ
. Π ΠΏΡΠΈΠΌΠ΅ΡΡ, ΡΡΠΎ ΠΌΠΎΠΆΠ΅Ρ ΠΊΠ°ΡΠ°ΡΡΡΡ Π°Π½Π°Π»ΠΈΠ·Π° ΡΠ°Π±ΠΎΡΡ Π΄ΠΈΡΠΊΠΎΠ²ΡΡ
ΠΌΠ°ΡΡΠΈΠ²ΠΎΠ², ΠΎΠ±Π»Π°ΡΠ½ΡΡ
Π²ΡΡΠΈΡΠ»Π΅Π½ΠΈΠΉ, Π²ΡΡΠΎΠΊΠΎΠΏΡΠΎΠΈΠ·Π²ΠΎΠ΄ΠΈΡΠ΅Π»ΡΠ½ΡΡ
ΡΠ΅ΡΠ²ΠΈΡΠΎΠ² ΠΈ Π΄Π°ΠΆΠ΅ ΠΏΡΠΎΡΠ΅ΡΡΠ°ΠΊΠΎΠΌΠΏΠ»Π΅ΠΊΡΠ°ΡΠΈΠΈ Π·Π°ΠΊΠ°Π·ΠΎΠ² Π½Π° ΡΠΊΠ»Π°Π΄Π΅. ΠΡΠ»ΠΈ Π² ΠΏΠ΅ΡΠ²ΠΎΠΉ ΡΠ°ΡΡΠΈ ΠΎΠ±Π·ΠΎΡΠ° Π±ΡΠ»ΠΈ ΠΎΠΏΠΈΡΠ°Π½Ρ ΠΎΡΠΎΠ±Π΅Π½Π½ΠΎΡΡΠΈΠΏΠΎΡΡΡΠΎΠ΅Π½ΠΈΡ Π΄Π°Π½Π½ΠΎΠΉ ΠΌΠΎΠ΄Π΅Π»ΠΈ ΠΈ ΡΠΎΠ΄ΡΡΠ²Π΅Π½Π½ΡΡ
Π΅ΠΉ ΡΠΈΡΡΠ΅ΠΌ, Π° ΡΠ°ΠΊΠΆΠ΅ ΠΏΡΠΈΠ²Π΅Π΄Π΅Π½ΠΎ ΠΏΠΎΠ΄ΡΠΎΠ±Π½ΠΎΠ΅ ΠΎΠΏΠΈΡΠ°Π½ΠΈΠ΅ ΠΏΠΎΠ΄Ρ
ΠΎΠ΄Π° ΠΊ ΠΏΠΎΠ»ΡΡΠ΅Π½ΠΈΡ ΡΠΎΡΠ½ΠΎΠ³ΠΎ Π²ΡΡΠ°ΠΆΠ΅Π½ΠΈΡ ΡΡΠ΅Π΄Π½Π΅Π³ΠΎ Π²ΡΠ΅ΠΌΠ΅Π½ΠΈ ΠΎΡΠΊΠ»ΠΈΠΊΠ° Π² ΡΠ»ΡΡΠ°Π΅ Π΄Π²ΡΡ
ΠΏΡΠΈΠ±ΠΎΡΠΎΠ² ΠΈ ΠΏΡΠ΅Π΄ΡΡΠ°Π²Π»Π΅Π½ΠΎ Π½Π΅ΡΠΊΠΎΠ»ΡΠΊΠΎ ΠΌΠ΅ΡΠΎΠ΄ΠΎΠ² ΠΏΡΠΈΠ±Π»ΠΈΠΆΠ΅Π½Π½ΠΎΠ³ΠΎ Π°Π½Π°Π»ΠΈΠ·Π° Π΄Π°Π½Π½ΠΎΠΉ Ρ
Π°ΡΠ°ΠΊΡΠ΅ΡΠΈΡΡΠΈΠΊΠΈ Π² ΡΠ»ΡΡΠ°Π΅, ΠΊΠΎΠ³Π΄Π° ΡΠΈΡΠ»ΠΎ ΠΏΡΠΈΠ±ΠΎΡΠΎΠ² Π±ΠΎΠ»ΡΡΠ΅ Π΄Π²ΡΡ
, ΡΠΎ Π²ΠΎ Π²ΡΠΎΡΠΎΠΉ ΡΠ°ΡΡΠΈ ΠΎΠ±Π·ΠΎΡΠ° ΠΏΡΠ΅Π΄ΡΡΠ°Π²Π»Π΅Π½ΠΎΠΎΠΏΠΈΡΠ°Π½ΠΈΠ΅ Π΄ΡΡΠ³ΠΈΡ
ΡΡΡΠ΅ΡΡΠ²ΡΡΡΠΈΡ
ΠΌΠ΅ΡΠΎΠ΄ΠΎΠ² Π°ΠΏΠΏΡΠΎΠΊΡΠΈΠΌΠ°ΡΠΈΠΈ ΡΡΠ΅Π΄Π½Π΅Π³ΠΎ Π²ΡΠ΅ΠΌΠ΅Π½ΠΈ ΠΎΡΠΊΠ»ΠΈΠΊΠ°. Π ΡΠ°ΡΡΠ½ΠΎΡΡΠΈ, ΠΊ ΡΠ°ΡΡΠΌΠ°ΡΡΠΈΠ²Π°Π΅ΠΌΡΠΌ ΠΏΠΎΠ΄Ρ
ΠΎΠ΄Π°ΠΌ ΠΏΡΠΈΠ±Π»ΠΈΠΆΠ΅Π½Π½ΠΎΠ³ΠΎ Π°Π½Π°Π»ΠΈΠ·Π° Π²ΡΠ΅ΠΌΠ΅Π½ΠΈ ΠΎΡΠΊΠ»ΠΈΠΊΠ° ΠΎΡΠ½ΠΎΡΡΡΡΡ:ΠΌΠ°ΡΡΠΈΡΠ½ΠΎ-Π³Π΅ΠΎΠΌΠ΅ΡΡΠΈΡΠ΅ΡΠΊΠΈΠΉ ΠΌΠ΅ΡΠΎΠ΄, Π°Π½Π°Π»ΠΈΠ· Ρ ΠΏΠΎΠΌΠΎΡΡΡ ΠΏΠΎΡΡΠ΄ΠΊΠΎΠ²ΡΡ
ΡΡΠ°ΡΠΈΡΡΠΈΠΊ Π΄Π»Ρ ΡΠ°Π·Π»ΠΈΡΠ½ΡΡ
ΡΠΈΠΏΠΎΠ² ΡΠ°ΡΠΏΡΠ΅Π΄Π΅Π»Π΅Π½ΠΈΡ Π²ΡΠ΅ΠΌΠ΅Π½ΠΈ ΠΏΡΠ΅Π±ΡΠ²Π°Π½ΠΈΡ ΠΏΠΎΠ΄Π·Π°ΠΏΡΠΎΡΠΎΠ²
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
A Taylor Series Approach for Service-Coupled Queueing Systems with Intermediate Load
This paper investigates the performance of a queueing model with multiple finite queues and a single server. Departures from the queues are synchronised or coupled which means that a service completion leads to a departure in every queue and that service is temporarily interrupted whenever any of the queues is empty. We focus on the numerical analysis of this queueing model in a Markovian setting: the arrivals in the different queues constitute Poisson processes and the service times are exponentially distributed. Taking into account the state space explosion problem associated with multidimensional Markov processes, we calculate the terms in the series expansion in the service rate of the stationary distribution of the Markov chain as well as various performance measures when the system is (i) overloaded and (ii) under intermediate load. Our numerical results reveal that, by calculating the series expansions of performance measures around a few service rates, we get accurate estimates of various performance measures once the load is above 40% to 50%