Diffusion Limit for Single-Server Retrial Queues with Renewal Input and Outgoing Calls
This paper studies a single-server retrial queue with two types of calls (incoming and outgoing calls). Incoming calls arrive at the server according to a renewal process, and outgoing calls of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline&q...
| Main Authors: | , , , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2022-03-01
|
| Series: | Mathematics |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2227-7390/10/6/948 |
| _version_ | 1797445405936451584 |
|---|---|
| author | Anatoly Nazarov Tuan Phung-Duc Svetlana Paul Olga Lizyura |
| author_facet | Anatoly Nazarov Tuan Phung-Duc Svetlana Paul Olga Lizyura |
| author_sort | Anatoly Nazarov |
| collection | DOAJ |
| description | This paper studies a single-server retrial queue with two types of calls (incoming and outgoing calls). Incoming calls arrive at the server according to a renewal process, and outgoing calls of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>N</mi><mo>−</mo><mn>1</mn></mrow></semantics></math></inline-formula> (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>N</mi><mo>≥</mo><mn>2</mn></mrow></semantics></math></inline-formula>) categories occur according to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>N</mi><mo>−</mo><mn>1</mn></mrow></semantics></math></inline-formula> independent Poisson processes. Upon arrival, if the server is occupied, an incoming call joins a virtual infinite queue called the orbit, and after an exponentially distributed time in orbit enters the server again, while outgoing calls are lost if the server is busy at the time of their arrivals. Although M/G/1 retrial queues and their variants are extensively studied in the literature, the GI/M/1 retrial queues are less studied due to their complexity. This paper aims to obtain a diffusion limit for the number of calls in orbit when the retrial rate is extremely low. Based on the diffusion limit, we built an approximation to the distribution of the number of calls in orbit. |
| first_indexed | 2024-03-09T13:25:24Z |
| format | Article |
| id | doaj.art-773c586c7dd1492e97fa0b2d1a6e484a |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-09T13:25:24Z |
| publishDate | 2022-03-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-773c586c7dd1492e97fa0b2d1a6e484a2023-11-30T21:24:31ZengMDPI AGMathematics2227-73902022-03-0110694810.3390/math10060948Diffusion Limit for Single-Server Retrial Queues with Renewal Input and Outgoing CallsAnatoly Nazarov0Tuan Phung-Duc1Svetlana Paul2Olga Lizyura3Institute of Applied Mathematics and Computer Science, National Research Tomsk State University, 36 Lenina Ave., 634050 Tomsk, RussiaDepartment of Policy and Planning Sciences, Faculty of Engineering, Information and Systems, University of Tsukuba, 1-1-1 Tennodai, Tsukuba 305-8573, Ibaraki, JapanInstitute of Applied Mathematics and Computer Science, National Research Tomsk State University, 36 Lenina Ave., 634050 Tomsk, RussiaInstitute of Applied Mathematics and Computer Science, National Research Tomsk State University, 36 Lenina Ave., 634050 Tomsk, RussiaThis paper studies a single-server retrial queue with two types of calls (incoming and outgoing calls). Incoming calls arrive at the server according to a renewal process, and outgoing calls of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>N</mi><mo>−</mo><mn>1</mn></mrow></semantics></math></inline-formula> (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>N</mi><mo>≥</mo><mn>2</mn></mrow></semantics></math></inline-formula>) categories occur according to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>N</mi><mo>−</mo><mn>1</mn></mrow></semantics></math></inline-formula> independent Poisson processes. Upon arrival, if the server is occupied, an incoming call joins a virtual infinite queue called the orbit, and after an exponentially distributed time in orbit enters the server again, while outgoing calls are lost if the server is busy at the time of their arrivals. Although M/G/1 retrial queues and their variants are extensively studied in the literature, the GI/M/1 retrial queues are less studied due to their complexity. This paper aims to obtain a diffusion limit for the number of calls in orbit when the retrial rate is extremely low. Based on the diffusion limit, we built an approximation to the distribution of the number of calls in orbit.https://www.mdpi.com/2227-7390/10/6/948retrial queuetwo-way communicationrenewal processdiffusion approximationincoming calloutgoing call |
| spellingShingle | Anatoly Nazarov Tuan Phung-Duc Svetlana Paul Olga Lizyura Diffusion Limit for Single-Server Retrial Queues with Renewal Input and Outgoing Calls Mathematics retrial queue two-way communication renewal process diffusion approximation incoming call outgoing call |
| title | Diffusion Limit for Single-Server Retrial Queues with Renewal Input and Outgoing Calls |
| title_full | Diffusion Limit for Single-Server Retrial Queues with Renewal Input and Outgoing Calls |
| title_fullStr | Diffusion Limit for Single-Server Retrial Queues with Renewal Input and Outgoing Calls |
| title_full_unstemmed | Diffusion Limit for Single-Server Retrial Queues with Renewal Input and Outgoing Calls |
| title_short | Diffusion Limit for Single-Server Retrial Queues with Renewal Input and Outgoing Calls |
| title_sort | diffusion limit for single server retrial queues with renewal input and outgoing calls |
| topic | retrial queue two-way communication renewal process diffusion approximation incoming call outgoing call |
| url | https://www.mdpi.com/2227-7390/10/6/948 |
| work_keys_str_mv | AT anatolynazarov diffusionlimitforsingleserverretrialqueueswithrenewalinputandoutgoingcalls AT tuanphungduc diffusionlimitforsingleserverretrialqueueswithrenewalinputandoutgoingcalls AT svetlanapaul diffusionlimitforsingleserverretrialqueueswithrenewalinputandoutgoingcalls AT olgalizyura diffusionlimitforsingleserverretrialqueueswithrenewalinputandoutgoingcalls |