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Queuing Theory

Queuing Theory is a field of mathematics concerned with queues (i.e. waiting lines). It is considered to be a branch of operations research, since valuable business decisions can be made based on the inputs from queuing theory. The fundamental problem which queuing theory attempts to solve is the prediction of the waiting line length and waiting time using a model-based approach. Thus, it can be also stated that it is concerned with explaining the queue formation and propagation. Queuing theory is closely related to probabilistic theory and stochastic processes in terms of using its valuable mathematical tools. On the other hand, methods from queuing theory are applicable in numerous disciplines: communications, operations research, reliability engineering, microprocessor electronics, management & business organization, computing, traffic engineering, psychology, etc. You can probably think of many other real-world applications where knowing how a waiting line will change is an advantage. Whether we wait for service to be performed (e.g. supermarkets, banks), wait for public transport to arrive (bus, subway), or any kind of response from a machine (computers or smartphones), queuing is part of our everyday experience.

Some of the topics covered in a standard course on queuing theory would be:

Topics from probability and stochastic processes relevant for queuing theory (random variables, random processes, types of distributions, moments, central limit theorem, law of large numbers, etc)
Performance measures of queuing systems
Description of queuing systems using Kendall’s notation
Infinite- and finite-source systems
Analysis of scheduling policies (first-come-first-served, first-come-last-served, served-in-random-order, processor sharing, shortest job first, etc)
Birth-death processes and simple Markovian queues
Multidimensional birth-death processes
Single- and multi-server queues
Queuing networks (Jackson networks, BCMP theorem)
Little’s law
Asymptotic analysis of queues
Programming simulations of queuing models
Renewal theory

To understand the basics of Queuing Theory, it is important to first have a brief overview of the standard notation:

λ: average rate of arrival of customer
µ: average service rate
ρ = λ / µ for single server queues: the probability that the server is busy or the probability that someone is being served
s: number of servers
P_n: probability that there are n customers in the system
L: average number of the customers present in the system

Additionally, models are described via Kendall’s notation as follows:

A / B / s / q / c / p

where:

A stands for the distribution describing arrival process,

B stands for the distribution describing service (distribution of service duration),

s stands for the total number of servers,

q stands for the queuing discipline (FIFO, LIFO, SIRO...)

c stands for the system capacity in case of a finite queue, and

p stands for the population size (number of possible customers).

When omitted, common practice is to assume that q = FIFO (First-In-First-Out), c = ∞ (i.e. infinite queue), and p = ∞ (i.e. open system). A typical example of a queuing model is M/M/1: M stands for Markovian process in both arrival and service process, with single (1) server.

Apart from your study materials and classes, you can find many useful lectures and various explanations concerning queuing theory on these websites:

MIT open course - Queues: Theory and Applications
Introduction to Queuing Theory from Florida Atlantic University by Dr. Bob Cooper: lecture videos and book
Basic Queuing Theory, Dr. Janos Sztrik, University of Debrecen (book)
Basic relations in Queuing Theory from Missouri University of Science and Technology
Fundamental formulas in Queuing Theory from KTH Royal Institute of Technology (Sweden)

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Illuminate\Cache\Events\CacheHit (549μs)
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Illuminate\Cache\Events\CacheHit (1.62ms)
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Illuminate\Database\Events\StatementPrepared (7.77ms)
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Illuminate\Database\Events\StatementPrepared (796μs)
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Illuminate\Database\Events\StatementPrepared (680μs)
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eloquent.retrieved: App\Models\Subject (96μs)
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Illuminate\Database\Events\QueryExecuted (988μs)
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creating: site.blocks.reviews (1ms)
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creating: site.blocks.submit-homework (144μs)
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creating: site.blocks.faq (115μs)
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eloquent.retrieved: App\Models\Twill\Slugs\TwillFaqSlug (108μs)
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creating: components.faq-item (158μs)
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creating: components.open-graph (266μs)
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Illuminate\Cache\Events\CacheHit (5.44ms)
Illuminate\Cache\Events\CacheHit (340μs)
Illuminate\Cache\Events\CacheHit (343μs)
Illuminate\Cache\Events\CacheHit (206μs)
Illuminate\Cache\Events\CacheHit (211μs)
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Illuminate\Cache\Events\CacheHit (229μs)
Illuminate\Cache\Events\CacheHit (239μs)
Illuminate\Cache\Events\CacheHit (222μs)
Illuminate\Cache\Events\CacheHit (193μs)
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creating: components.forms.free-tool-download (104μs)
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creating: components.forms.claim-free-worksheet (100μs)
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creating: components.forms.tutor-subscription-waitlist (111μs)
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creating: components.forms.tutor-subscription-join (101μs)
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creating: components.forms.tutor-support (98μs)
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513 x Illuminate\Cache\Events\CacheHit (23.27%)	295ms
35 x Illuminate\Database\Events\QueryExecuted (7.62%)	96.64ms
35 x Illuminate\Database\Events\StatementPrepared (7.17%)	90.90ms
12 x Illuminate\Cache\Events\CacheMissed (2.13%)	27.00ms
12 x Illuminate\Cache\Events\KeyWritten (1.51%)	19.18ms
6 x creating: components.tutors-card (1.31%)	16.67ms
1 x Illuminate\Foundation\Events\LocaleUpdated (0.96%)	12.21ms
1 x Illuminate\Routing\Events\Routing (0.9%)	11.48ms
15 x eloquent.retrieved: App\Models\SubjectCat (0.22%)	2.75ms
2 x Illuminate\Database\Events\ConnectionEstablished (0.14%)	1.82ms
1 x creating: site.blocks.reviews (0.08%)	1.00ms
1 x creating: site.blocks.subject-homework (0.07%)	875μs
45 x eloquent.retrieved: App\Models\Subject (0.07%)	853μs
4 x eloquent.retrieved: App\Models\Twill\TwillFaq (0.05%)	601μs
4 x creating: components.faq-item (0.04%)	563μs
1 x Illuminate\Routing\Events\RouteMatched (0.04%)	476μs
6 x composing: components.tutors-card (0.03%)	435μs
6 x eloquent.retrieved: App\Models\Twill\Slugs\TwillFaqSlug (0.03%)	408μs
1 x eloquent.booting: App\Models\User\TutorAttributes\TutorDegree (0.03%)	371μs
1 x creating: components.canonical (0.03%)	358μs
1 x creating: site.blocks.text-with-image (0.03%)	344μs
1 x creating: site.blocks.how-it-works (0.03%)	320μs
1 x creating: components.homework-blocks (0.02%)	291μs
4 x composing: components.faq-item (0.02%)	291μs
1 x creating: site.headers.header (0.02%)	281μs
1 x creating: components.open-graph (0.02%)	266μs
1 x creating: components.forms.contact-us (0.02%)	263μs
6 x eloquent.retrieved: App\Models\HomeworkLibrary\HomeworkLibrary (0.02%)	246μs
1 x eloquent.booted: App\Models\Twill\TwillFaq (0.02%)	238μs
1 x creating: site.blocks.work-with-tutors (0.02%)	226μs
1 x eloquent.booting: App\Models\Subject (0.02%)	224μs
6 x eloquent.retrieved: App\Models\User\Tutor (0.02%)	224μs
1 x composing: components.forms.contact-us (0.02%)	222μs
1 x creating: components.breadcrumbs (0.02%)	220μs
1 x composing: components.footer (0.02%)	216μs
1 x eloquent.booting: App\Models\Twill\Slugs\TwillFaqSlug (0.02%)	214μs
1 x creating: site.layouts.app (0.02%)	213μs
1 x eloquent.booting: App\Models\User\Tutor (0.02%)	213μs
1 x eloquent.booting: App\Models\Twill\TwillFaq (0.02%)	211μs
1 x eloquent.booting: App\Models\Meta (0.02%)	209μs
1 x creating: components.tutors-cards (0.02%)	202μs
1 x composing: components.homework-blocks (0.01%)	190μs
1 x eloquent.booted: App\Models\User\Tutor (0.01%)	181μs
1 x creating: components.forms.get-started (0.01%)	176μs
1 x creating: components.footer (0.01%)	157μs
1 x eloquent.retrieved: App\Models\Meta (0.01%)	152μs
1 x eloquent.booting: App\Models\HomeworkLibrary\HomeworkLibrary (0.01%)	151μs
1 x creating: site.blocks.submit-homework (0.01%)	144μs
1 x eloquent.booted: App\Models\HomeworkLibrary\HomeworkLibrary (0.01%)	132μs
1 x composing: components.forms.claim-free-worksheet (0.01%)	122μs
1 x creating: site.blocks.subject-tutors (0.01%)	118μs
1 x creating: site.blocks.faq (0.01%)	115μs
1 x composing: components.breadcrumbs (0.01%)	115μs
1 x creating: components.forms.tutor-subscription-waitlist (0.01%)	111μs
1 x creating: components.forms.free-tool-download (0.01%)	104μs
1 x creating: subjects.show (0.01%)	101μs
1 x creating: components.forms.tutor-subscription-join (0.01%)	101μs
1 x composing: subjects.show (0.01%)	100μs
1 x creating: components.forms.claim-free-worksheet (0.01%)	100μs
1 x composing: components.tutors-cards (0.01%)	98μs
1 x creating: components.forms.tutor-support (0.01%)	98μs
1 x composing: components.canonical (0.01%)	96μs
1 x composing: components.forms.get-started (0.01%)	88μs
1 x composing: components.open-graph (0.01%)	84μs
1 x eloquent.booted: App\Models\Twill\Slugs\TwillFaqSlug (0.01%)	73μs
1 x eloquent.booting: App\Models\SubjectCat (0.01%)	69μs
1 x composing: components.forms.free-tool-download (0.01%)	68μs
1 x composing: components.forms.tutor-subscription-waitlist (0.01%)	67μs
1 x composing: components.forms.tutor-subscription-join (0.01%)	67μs
1 x eloquent.booted: App\Models\Subject (0.01%)	66μs
1 x composing: components.forms.tutor-support (0.01%)	64μs
1 x eloquent.booted: App\Models\User\TutorAttributes\TutorDegree (0%)	57μs
1 x eloquent.booted: App\Models\Meta (0%)	56μs
1 x composing: site.headers.header (0%)	56μs
1 x eloquent.booted: App\Models\SubjectCat (0%)	45μs
1 x composing: site.blocks.text-with-image (0%)	39μs
1 x composing: site.blocks.reviews (0%)	29μs
1 x composing: site.blocks.subject-homework (0%)	26μs
1 x composing: site.blocks.how-it-works (0%)	22μs
1 x composing: site.blocks.submit-homework (0%)	21μs
1 x composing: site.blocks.work-with-tutors (0%)	21μs
1 x composing: site.blocks.subject-tutors (0%)	20μs
1 x composing: site.layouts.app (0%)	20μs
1 x composing: site.blocks.faq (0%)	18μs

34 templates were rendered

1x subjects.showshow.blade.phpblade
1x components.breadcrumbsbreadcrumbs.blade.phpblade
1x site.blocks.how-it-workshow-it-works.blade.phpblade
1x site.blocks.subject-tutorssubject-tutors.blade.phpblade
1x components.tutors-cardstutors-cards.blade.phpblade
6x components.tutors-cardtutors-card.blade.phpblade
1x site.blocks.subject-homeworksubject-homework.blade.phpblade
1x components.homework-blockshomework-blocks.blade.phpblade
1x site.blocks.reviewsreviews.blade.phpblade
1x site.blocks.submit-homeworksubmit-homework.blade.phpblade
1x site.blocks.faqfaq.blade.phpblade
4x components.faq-itemfaq-item.blade.phpblade
1x site.blocks.text-with-imagetext-with-image.blade.phpblade
1x site.blocks.work-with-tutorswork-with-tutors.blade.phpblade
1x site.layouts.appapp.blade.phpblade
1x components.canonicalcanonical.blade.phpblade
1x components.open-graphopen-graph.blade.phpblade
1x site.headers.headerheader.blade.phpblade
1x components.footerfooter.blade.phpblade
1x components.forms.contact-uscontact-us.blade.phpblade
1x components.forms.get-startedget-started.blade.phpblade
1x components.forms.free-tool-downloadfree-tool-download.blade.phpblade
1x components.forms.claim-free-worksheetclaim-free-worksheet.blade.phpblade
1x components.forms.tutor-subscription-waitlisttutor-subscription-waitlist.blade.phpblade
1x components.forms.tutor-subscription-jointutor-subscription-join.blade.phpblade
1x components.forms.tutor-supporttutor-support.blade.phpblade

uri: GET subjects/{categoryUri}/{subjectUri?}
middleware: web, utm.parameters
controller: App\Http\Controllers\SubjectController@show
namespace
where
as: subjects.show
file: app/Http/Controllers/SubjectController.php:43-96

35 statements were executed, 2 of which were duplicates, 33 unique. Show only duplicated149ms

Connection Establishedtwenty4_siteSubjectRepository.php#123

Backtrace

13. app/Repositories/SubjectRepository.php:123
14. app/Http/Controllers/SubjectController.php:54
15. vendor/laravel/framework/src/Illuminate/Routing/Controller.php:54
16. vendor/laravel/framework/src/Illuminate/Routing/ControllerDispatcher.php:43
17. vendor/laravel/framework/src/Illuminate/Routing/Route.php:260

select * from `subjects` where `urlFriendlyTitle` = 'Queuing-Theory' limit 1

6.76mstwenty4_siteSubjectRepository.php#123

Bindings	0: Queuing-Theory
Backtrace	15. app/Repositories/SubjectRepository.php:123 16. app/Http/Controllers/SubjectController.php:54 17. vendor/laravel/framework/src/Illuminate/Routing/Controller.php:54 18. vendor/laravel/framework/src/Illuminate/Routing/ControllerDispatcher.php:43 19. vendor/laravel/framework/src/Illuminate/Routing/Route.php:260

select * from `subject_cats` where `subject_cats`.`id` in (1)

990μstwenty4_siteSubjectRepository.php#123

Backtrace

20. app/Repositories/SubjectRepository.php:123
21. app/Http/Controllers/SubjectController.php:54
22. vendor/laravel/framework/src/Illuminate/Routing/Controller.php:54
23. vendor/laravel/framework/src/Illuminate/Routing/ControllerDispatcher.php:43
24. vendor/laravel/framework/src/Illuminate/Routing/Route.php:260

select * from `subject_cats` where `subject_cats`.`id` = 1 limit 1

9.78mstwenty4_siteSubject.php#100

Bindings	0: 1
Backtrace	18. app/Models/Subject.php:100 19. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397 20. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419 22. app/Models/Subject.php:101 28. app/Http/Controllers/SubjectController.php:86

select * from `meta` where `meta`.`id` = 3 limit 1

950μstwenty4_siteMetaService.php#118

Bindings	0: 3
Backtrace	19. app/Services/MetaService.php:118 20. app/Services/MetaService.php:108 21. app/Services/MetaService.php:81 22. app/Http/Controllers/SubjectController.php:89 23. vendor/laravel/framework/src/Illuminate/Routing/Controller.php:54

Connection Establishedtwenty4_siteTutorRepository.php#40

Backtrace

13. app/Repositories/TutorRepository.php:40
14. app/View/Components/TutorsCards.php:28
15. app/View/Components/TutorsCards.php:22
17. vendor/laravel/framework/src/Illuminate/Container/Container.php:929
18. vendor/laravel/framework/src/Illuminate/Container/Container.php:770

select count(*) as aggregate from `users` where (`subjectsQualified` like '%|71|%' or `DSF_subjectsQualified` like '%|71|%' or `allsubjects` = 1) and `disable_profile` = 0 and `picture` != '' and `solutionrequestQualified` = 1 and `is_suspended` = 0 and `is_deactivated` = 0 and `status` != '' and `status` = 'tutor'

15.56mstwenty4_siteTutorsCards.php#35

Bindings	0: %\|71\|% 1: %\|71\|% 2: 1 3: 0 4: 5: 1 6: 0 7: 0 8: 9: tutor
Backtrace	15. app/View/Components/TutorsCards.php:35 16. app/View/Components/TutorsCards.php:22 18. vendor/laravel/framework/src/Illuminate/Container/Container.php:929 19. vendor/laravel/framework/src/Illuminate/Container/Container.php:770 20. vendor/laravel/framework/src/Illuminate/Foundation/Application.php:933

select * from `users` where (`subjectsQualified` like '%|71|%' or `DSF_subjectsQualified` like '%|71|%' or `allsubjects` = 1) and `disable_profile` = 0 and `picture` != '' and `solutionrequestQualified` = 1 and `is_suspended` = 0 and `is_deactivated` = 0 and `status` != '' and `status` = 'tutor' order by `completed_requests` desc limit 6 offset 0

16.31mstwenty4_siteTutorsCards.php#35

Bindings	0: %\|71\|% 1: %\|71\|% 2: 1 3: 0 4: 5: 1 6: 0 7: 0 8: 9: tutor
Backtrace	15. app/View/Components/TutorsCards.php:35 16. app/View/Components/TutorsCards.php:22 18. vendor/laravel/framework/src/Illuminate/Container/Container.php:929 19. vendor/laravel/framework/src/Illuminate/Container/Container.php:770 20. vendor/laravel/framework/src/Illuminate/Foundation/Application.php:933

select * from `tutor_degrees` where `tutor_degrees`.`tutor_id` = 2821 and `tutor_degrees`.`tutor_id` is not null order by `year` desc limit 1

5.05mstwenty4_siteTutor.php#361

Bindings	0: 2821
Backtrace	18. app/Models/User/Tutor.php:361 19. view::components.tutors-card:11 21. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110 22. vendor/laravel/framework/src/Illuminate/View/Engines/PhpEngine.php:58 23. vendor/livewire/livewire/src/ComponentConcerns/RendersLivewireComponents.php:69

select * from `subject_cats` where `subject_cats`.`id` = 3 limit 1

7.24mstwenty4_siteSubject.php#100

Bindings	0: 3
Backtrace	18. app/Models/Subject.php:100 19. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397 20. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419 22. app/Models/Subject.php:101 34. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110

select * from `subject_cats` where `subject_cats`.`id` = 5 limit 1

970μstwenty4_siteSubject.php#100

Bindings	0: 5
Backtrace	18. app/Models/Subject.php:100 19. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397 20. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419 22. app/Models/Subject.php:101 34. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110

select * from `subject_cats` where `subject_cats`.`id` = 34 limit 1

6.2mstwenty4_siteSubject.php#100

Bindings	0: 34
Backtrace	18. app/Models/Subject.php:100 19. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397 20. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419 22. app/Models/Subject.php:101 34. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110

select * from `subject_cats` where `subject_cats`.`id` = 10 limit 1

1.83mstwenty4_siteSubject.php#100

Bindings	0: 10
Backtrace	18. app/Models/Subject.php:100 19. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397 20. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419 22. app/Models/Subject.php:101 34. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110

select * from `tutor_degrees` where `tutor_degrees`.`tutor_id` = 6531 and `tutor_degrees`.`tutor_id` is not null order by `year` desc limit 1

940μstwenty4_siteTutor.php#361

Bindings	0: 6531
Backtrace	18. app/Models/User/Tutor.php:361 19. view::components.tutors-card:11 21. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110 22. vendor/laravel/framework/src/Illuminate/View/Engines/PhpEngine.php:58 23. vendor/livewire/livewire/src/ComponentConcerns/RendersLivewireComponents.php:69

select * from `subject_cats` where `subject_cats`.`id` = 31 limit 1

3.59mstwenty4_siteSubject.php#100

Bindings	0: 31
Backtrace	18. app/Models/Subject.php:100 19. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397 20. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419 22. app/Models/Subject.php:101 34. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110

select * from `tutor_degrees` where `tutor_degrees`.`tutor_id` = 76351 and `tutor_degrees`.`tutor_id` is not null order by `year` desc limit 1

1.66mstwenty4_siteTutor.php#361

Bindings	0: 76351
Backtrace	18. app/Models/User/Tutor.php:361 19. view::components.tutors-card:11 21. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110 22. vendor/laravel/framework/src/Illuminate/View/Engines/PhpEngine.php:58 23. vendor/livewire/livewire/src/ComponentConcerns/RendersLivewireComponents.php:69

select * from `subject_cats` where `subject_cats`.`id` = 36 limit 1

1.16mstwenty4_siteSubject.php#100

Bindings	0: 36
Backtrace	18. app/Models/Subject.php:100 19. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397 20. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419 22. app/Models/Subject.php:101 34. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110

select * from `subject_cats` where `subject_cats`.`id` = 42 limit 1

5.97mstwenty4_siteSubject.php#100

Bindings	0: 42
Backtrace	18. app/Models/Subject.php:100 19. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397 20. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419 22. app/Models/Subject.php:101 34. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110

select * from `tutor_degrees` where `tutor_degrees`.`tutor_id` = 76533 and `tutor_degrees`.`tutor_id` is not null order by `year` desc limit 1

1.66mstwenty4_siteTutor.php#361

Bindings	0: 76533
Backtrace	18. app/Models/User/Tutor.php:361 19. view::components.tutors-card:11 21. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110 22. vendor/laravel/framework/src/Illuminate/View/Engines/PhpEngine.php:58 23. vendor/livewire/livewire/src/ComponentConcerns/RendersLivewireComponents.php:69

select * from `tutor_degrees` where `tutor_degrees`.`tutor_id` = 104169 and `tutor_degrees`.`tutor_id` is not null order by `year` desc limit 1

14.49mstwenty4_siteTutor.php#361

Bindings	0: 104169
Backtrace	18. app/Models/User/Tutor.php:361 19. view::components.tutors-card:11 21. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110 22. vendor/laravel/framework/src/Illuminate/View/Engines/PhpEngine.php:58 23. vendor/livewire/livewire/src/ComponentConcerns/RendersLivewireComponents.php:69

select `subjects`.*, `subject_cats`.`title` as `category_title`, `subject_cats`.`id` as `category_id` from `subjects` inner join `subject_cats` on `subject_cats`.`id` = `subjects`.`catID` where `subjects`.`id` in (19, 324, 34, 134, 2, 133, 1, 127, 257, 7, 66, 60, 68, 73, 41, 8, 71, 67, 299, 4)

4.56mstwenty4_siteTutor.php#214

Bindings	0: 19 1: 324 2: 34 3: 134 4: 2 5: 133 6: 1 7: 127 8: 257 9: 7 10: 66 11: 60 12: 68 13: 73 14: 41 15: 8 16: 71 17: 67 18: 299 19: 4
Backtrace	14. app/Models/User/Tutor.php:214 15. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397 16. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419 18. app/Models/User/Tutor.php:216 24. app/Models/User/Tutor.php:267

select * from `subject_cats` where `subject_cats`.`id` in (1, 4)

1.17mstwenty4_siteTutor.php#214

Backtrace

19. app/Models/User/Tutor.php:214
20. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
21. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
23. app/Models/User/Tutor.php:216
29. app/Models/User/Tutor.php:267

select * from `tutor_degrees` where `tutor_degrees`.`tutor_id` = 115638 and `tutor_degrees`.`tutor_id` is not null order by `year` desc limit 1

1.04mstwenty4_siteTutor.php#361

Bindings	0: 115638
Backtrace	18. app/Models/User/Tutor.php:361 19. view::components.tutors-card:11 21. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110 22. vendor/laravel/framework/src/Illuminate/View/Engines/PhpEngine.php:58 23. vendor/livewire/livewire/src/ComponentConcerns/RendersLivewireComponents.php:69

select `subjects`.*, `subject_cats`.`title` as `category_title`, `subject_cats`.`id` as `category_id` from `subjects` inner join `subject_cats` on `subject_cats`.`id` = `subjects`.`catID` where `subjects`.`id` in (324, 512, 245, 34, 134, 2, 133, 1, 6, 127, 10, 7, 349, 9, 60, 125, 41, 8, 71, 67, 48, 299, 259)

4.86mstwenty4_siteTutor.php#214

Bindings	0: 324 1: 512 2: 245 3: 34 4: 134 5: 2 6: 133 7: 1 8: 6 9: 127 10: 10 11: 7 12: 349 13: 9 14: 60 15: 125 16: 41 17: 8 18: 71 19: 67 20: 48 21: 299 22: 259
Backtrace	14. app/Models/User/Tutor.php:214 15. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397 16. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419 18. app/Models/User/Tutor.php:216 24. app/Models/User/Tutor.php:267

select * from `subject_cats` where `subject_cats`.`id` in (1, 3, 4)

2.81mstwenty4_siteTutor.php#214

Backtrace

19. app/Models/User/Tutor.php:214
20. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
21. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
23. app/Models/User/Tutor.php:216
29. app/Models/User/Tutor.php:267

select count(*) as aggregate from `solutionslibrary` where exists (select * from `subjects` where `solutionslibrary`.`subject` = `subjects`.`id`) and `subject` = 71 and `status` = 'published' and `price` > 0

7.74mstwenty4_siteHomeworkBlocks.php#27

Bindings	0: 71 1: published 2: 0
Backtrace	15. app/View/Components/HomeworkBlocks.php:27 16. vendor/laravel/framework/src/Illuminate/View/Component.php:102 17. view::site.blocks.subject-homework:4 19. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110 20. vendor/laravel/framework/src/Illuminate/View/Engines/PhpEngine.php:58

select * from `solutionslibrary` where exists (select * from `subjects` where `solutionslibrary`.`subject` = `subjects`.`id`) and `subject` = 71 and `status` = 'published' and `price` > 0 order by `purchase_count` desc limit 6 offset 0

1.73mstwenty4_siteHomeworkBlocks.php#27

Bindings	0: 71 1: published 2: 0
Backtrace	15. app/View/Components/HomeworkBlocks.php:27 16. vendor/laravel/framework/src/Illuminate/View/Component.php:102 17. view::site.blocks.subject-homework:4 19. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110 20. vendor/laravel/framework/src/Illuminate/View/Engines/PhpEngine.php:58

select * from `subjects` where `subjects`.`id` in (71)

1.05mstwenty4_siteHomeworkBlocks.php#27

Backtrace

20. app/View/Components/HomeworkBlocks.php:27
21. vendor/laravel/framework/src/Illuminate/View/Component.php:102
22. view::site.blocks.subject-homework:4
24. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
25. vendor/laravel/framework/src/Illuminate/View/Engines/PhpEngine.php:58

select * from `subject_cats` where `subject_cats`.`id` in (1)

1.06mstwenty4_siteHomeworkBlocks.php#27

Backtrace

25. app/View/Components/HomeworkBlocks.php:27
26. vendor/laravel/framework/src/Illuminate/View/Component.php:102
27. view::site.blocks.subject-homework:4
29. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
30. vendor/laravel/framework/src/Illuminate/View/Engines/PhpEngine.php:58

select * from `twill_faqs` where exists (select * from `twill_faq_slugs` where `twill_faqs`.`id` = `twill_faq_slugs`.`twill_faq_id` and `slug` = 'can-you-help-me-with-my-homework-in-less-than-24-hours' and `active` = 1 and `locale` = 'en' and `twill_faq_slugs`.`deleted_at` is null) and `twill_faqs`.`deleted_at` is null limit 1

9.51mstwenty4_siteFaqItem.php#15

Bindings	0: can-you-help-me-with-my-homework-in-less-than-24-hours 1: 1 2: en
Backtrace	15. app/View/Components/FaqItem.php:15 16. vendor/laravel/framework/src/Illuminate/View/Component.php:102 17. view::site.blocks.faq:7 19. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110 20. vendor/laravel/framework/src/Illuminate/View/Engines/PhpEngine.php:58

select * from `twill_faq_slugs` where `twill_faq_slugs`.`twill_faq_id` in (1) and `twill_faq_slugs`.`deleted_at` is null

1.02mstwenty4_siteFaqItem.php#15

Backtrace

20. app/View/Components/FaqItem.php:15
21. vendor/laravel/framework/src/Illuminate/View/Component.php:102
22. view::site.blocks.faq:7
24. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
25. vendor/laravel/framework/src/Illuminate/View/Engines/PhpEngine.php:58

select * from `twill_faqs` where exists (select * from `twill_faq_slugs` where `twill_faqs`.`id` = `twill_faq_slugs`.`twill_faq_id` and `slug` = 'can-you-help-me-with-my-examquiztest' and `active` = 1 and `locale` = 'en' and `twill_faq_slugs`.`deleted_at` is null) and `twill_faqs`.`deleted_at` is null limit 1

900μstwenty4_siteFaqItem.php#15

Bindings	0: can-you-help-me-with-my-examquiztest 1: 1 2: en
Backtrace	15. app/View/Components/FaqItem.php:15 16. vendor/laravel/framework/src/Illuminate/View/Component.php:102 17. view::site.blocks.faq:24 19. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110 20. vendor/laravel/framework/src/Illuminate/View/Engines/PhpEngine.php:58

select * from `twill_faq_slugs` where `twill_faq_slugs`.`twill_faq_id` in (2) and `twill_faq_slugs`.`deleted_at` is null

910μstwenty4_siteFaqItem.php#15

Backtrace

20. app/View/Components/FaqItem.php:15
21. vendor/laravel/framework/src/Illuminate/View/Component.php:102
22. view::site.blocks.faq:24
24. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
25. vendor/laravel/framework/src/Illuminate/View/Engines/PhpEngine.php:58

select * from `twill_faqs` where exists (select * from `twill_faq_slugs` where `twill_faqs`.`id` = `twill_faq_slugs`.`twill_faq_id` and `slug` = 'how-much-will-it-cost' and `active` = 1 and `locale` = 'en' and `twill_faq_slugs`.`deleted_at` is null) and `twill_faqs`.`deleted_at` is null limit 1

870μstwenty4_siteFaqItem.php#15

Bindings	0: how-much-will-it-cost 1: 1 2: en
Backtrace	15. app/View/Components/FaqItem.php:15 16. vendor/laravel/framework/src/Illuminate/View/Component.php:102 17. view::site.blocks.faq:41 19. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110 20. vendor/laravel/framework/src/Illuminate/View/Engines/PhpEngine.php:58

select * from `twill_faq_slugs` where `twill_faq_slugs`.`twill_faq_id` in (3) and `twill_faq_slugs`.`deleted_at` is null

1.06mstwenty4_siteFaqItem.php#15

Backtrace

20. app/View/Components/FaqItem.php:15
21. vendor/laravel/framework/src/Illuminate/View/Component.php:102
22. view::site.blocks.faq:41
24. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
25. vendor/laravel/framework/src/Illuminate/View/Engines/PhpEngine.php:58

select * from `twill_faqs` where exists (select * from `twill_faq_slugs` where `twill_faqs`.`id` = `twill_faq_slugs`.`twill_faq_id` and `slug` = 'what-kind-of-payments-do-you-accept' and `active` = 1 and `locale` = 'en' and `twill_faq_slugs`.`deleted_at` is null) and `twill_faqs`.`deleted_at` is null limit 1

6.89mstwenty4_siteFaqItem.php#15

Bindings	0: what-kind-of-payments-do-you-accept 1: 1 2: en
Backtrace	15. app/View/Components/FaqItem.php:15 16. vendor/laravel/framework/src/Illuminate/View/Component.php:102 17. view::site.blocks.faq:58 19. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110 20. vendor/laravel/framework/src/Illuminate/View/Engines/PhpEngine.php:58

select * from `twill_faq_slugs` where `twill_faq_slugs`.`twill_faq_id` in (4) and `twill_faq_slugs`.`deleted_at` is null

850μstwenty4_siteFaqItem.php#15

Backtrace

20. app/View/Components/FaqItem.php:15
21. vendor/laravel/framework/src/Illuminate/View/Component.php:102
22. view::site.blocks.faq:58
24. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
25. vendor/laravel/framework/src/Illuminate/View/Engines/PhpEngine.php:58

App\Models\Subject: 45Subject.php
App\Models\SubjectCat: 15SubjectCat.php
App\Models\User\Tutor: 6Tutor.php
App\Models\HomeworkLibrary\HomeworkLibrary: 6HomeworkLibrary.php
App\Models\Twill\Slugs\TwillFaqSlug: 6TwillFaqSlug.php
App\Models\Twill\TwillFaq: 4TwillFaq.php
App\Models\Meta: 1Meta.php

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