Question
A queuing system with a Poisson arrival rate and exponential service time has three servers, an average arrival rate of 7.5 customers per hour, and an average service time of 20 minutes per customer.
Answer the following questions. Explain ALL formulas and calculations used in your response.
a. What is the minimum number of queues required to avoid an overloaded system?
b. How many queues are required to ensure the utilization is less than 50%
c. If the system had a single queue, what would be required to ensure that the queue length is not at risk of approaching infinity?

1. Clarke’s Shoe Factory has an annual demand of 5,000 for a particular shoe style. One of their suppliers made an irresistible offer on shoelaces for the popular shoe. Shoelaces usually cost 10 cents each; however, if Clarke’s buy 10,000 laces in each order for the upcoming year, they will pay only 5 cents each; and if they buy 50,000 in each order, you will pay only 3 cents each. It costs $15 to place an order. Carrying cost for each shoelace is 20% of purchase price.
a. Showing working, determine which of these two quantities represents the most optimal order quantity?
b. Explain the response you gave justifying ALL formula selections and calculations.
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      • Illuminate\Cache\Events\CacheHit (198μs)
      • Illuminate\Cache\Events\CacheHit (163μs)
      • Illuminate\Cache\Events\CacheHit (179μs)
      • Illuminate\Cache\Events\CacheHit (676μs)
      • Illuminate\Cache\Events\CacheHit (239μs)
      • Illuminate\Cache\Events\CacheHit (239μs)
      • Illuminate\Cache\Events\CacheHit (240μs)
      • Illuminate\Cache\Events\CacheHit (567μs)
      • Illuminate\Cache\Events\CacheHit (194μs)
      • Illuminate\Cache\Events\CacheHit (171μs)
      • Illuminate\Cache\Events\CacheHit (239μs)
      • Illuminate\Cache\Events\CacheHit (232μs)
      • Illuminate\Cache\Events\CacheHit (192μs)
      • Illuminate\Cache\Events\CacheHit (244μs)
      • Illuminate\Cache\Events\CacheHit (252μs)
      • Illuminate\Cache\Events\CacheHit (164μs)
      • creating: site.layouts.app (781μs)
      • composing: site.layouts.app (30μs)
      • creating: components.canonical (652μs)
      • composing: components.canonical (116μs)
      • creating: components.open-graph (313μs)
      • composing: components.open-graph (92μs)
      • creating: site.headers.header (705μs)
      • composing: site.headers.header (237μs)
      • Illuminate\Cache\Events\CacheHit (15.71ms)
      • creating: components.footer (132μs)
      • composing: components.footer (183μs)
      • Illuminate\Cache\Events\CacheHit (1.41ms)
      • Illuminate\Cache\Events\CacheMissed (312μs)
      • Illuminate\Database\Events\StatementPrepared (5.37ms)
      • Illuminate\Database\Events\QueryExecuted (1.36ms)
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      • Illuminate\Cache\Events\KeyWritten (726μs)
      • Illuminate\Cache\Events\CacheHit (1.19ms)
      • Illuminate\Cache\Events\CacheHit (324μs)
      • Illuminate\Cache\Events\CacheHit (197μs)
      • Illuminate\Cache\Events\CacheHit (364μs)
      • Illuminate\Cache\Events\CacheHit (289μs)
      • Illuminate\Cache\Events\CacheHit (10.93ms)
      • Illuminate\Cache\Events\CacheHit (406μs)
      • Illuminate\Cache\Events\CacheHit (307μs)
      • Illuminate\Cache\Events\CacheHit (212μs)
      • Illuminate\Cache\Events\CacheHit (264μs)
      • Illuminate\Cache\Events\CacheHit (188μs)
      • Illuminate\Cache\Events\CacheHit (181μs)
      • creating: components.forms.contact-us (375μs)
      • composing: components.forms.contact-us (173μs)
      • creating: components.forms.get-started (139μs)
      • composing: components.forms.get-started (74μs)
      • creating: components.forms.free-tool-download (106μs)
      • composing: components.forms.free-tool-download (70μs)
      • creating: components.forms.claim-free-worksheet (210μs)
      • composing: components.forms.claim-free-worksheet (83μs)
      • creating: components.forms.tutor-subscription-waitlist (118μs)
      • composing: components.forms.tutor-subscription-waitlist (138μs)
      • creating: components.forms.tutor-subscription-join (115μs)
      • composing: components.forms.tutor-subscription-join (71μs)
      • creating: components.forms.tutor-support (103μs)
      • composing: components.forms.tutor-support (77μs)
      • 311 x Illuminate\Cache\Events\CacheHit (26.75%)
        217ms
        20 x Illuminate\Database\Events\StatementPrepared (9.48%)
        76.80ms
        20 x Illuminate\Database\Events\QueryExecuted (4.97%)
        40.29ms
        11 x Illuminate\Cache\Events\CacheMissed (1.23%)
        9.97ms
        1 x Illuminate\Routing\Events\RouteMatched (1.01%)
        8.21ms
        11 x Illuminate\Cache\Events\KeyWritten (0.86%)
        6.96ms
        1 x Illuminate\Foundation\Events\LocaleUpdated (0.69%)
        5.61ms
        12 x eloquent.retrieved: App\Models\SubjectCat (0.19%)
        1.56ms
        1 x Illuminate\Routing\Events\Routing (0.16%)
        1.34ms
        1 x Illuminate\Database\Events\ConnectionEstablished (0.16%)
        1.28ms
        1 x creating: site.layouts.app (0.1%)
        781μs
        1 x eloquent.booted: App\Models\HomeworkLibrary\HomeworkLibrary (0.09%)
        737μs
        1 x creating: site.headers.header (0.09%)
        705μs
        1 x eloquent.booting: App\Models\SubjectCat (0.08%)
        667μs
        1 x creating: components.canonical (0.08%)
        652μs
        1 x creating: components.forms.contact-us (0.05%)
        375μs
        1 x creating: homework.show (0.04%)
        342μs
        1 x creating: components.breadcrumbs (0.04%)
        328μs
        1 x creating: components.open-graph (0.04%)
        313μs
        2 x eloquent.retrieved: App\Models\Subject (0.04%)
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        6 x eloquent.retrieved: App\Models\HomeworkLibrary\HomeworkLibrary (0.04%)
        291μs
        1 x eloquent.booting: App\Models\HomeworkLibrary\HomeworkLibrary (0.03%)
        280μs
        5 x eloquent.retrieved: App\Models\HomeworkLibrary\HomeworkLibraryFile (0.03%)
        275μs
        1 x composing: components.breadcrumbs (0.03%)
        247μs
        1 x composing: site.headers.header (0.03%)
        237μs
        1 x creating: components.forms.claim-free-worksheet (0.03%)
        210μs
        1 x composing: components.footer (0.02%)
        183μs
        1 x composing: components.forms.contact-us (0.02%)
        173μs
        1 x eloquent.booting: App\Models\HomeworkLibrary\HomeworkLibraryFile (0.02%)
        165μs
        1 x creating: components.forms.get-started (0.02%)
        139μs
        1 x composing: homework.show (0.02%)
        138μs
        1 x composing: components.forms.tutor-subscription-waitlist (0.02%)
        138μs
        1 x creating: components.footer (0.02%)
        132μs
        1 x eloquent.booting: App\Models\Subject (0.02%)
        126μs
        1 x creating: components.forms.tutor-subscription-waitlist (0.01%)
        118μs
        1 x composing: components.canonical (0.01%)
        116μs
        1 x creating: components.forms.tutor-subscription-join (0.01%)
        115μs
        1 x creating: components.forms.free-tool-download (0.01%)
        106μs
        1 x creating: components.forms.tutor-support (0.01%)
        103μs
        1 x composing: components.open-graph (0.01%)
        92μs
        1 x composing: components.forms.claim-free-worksheet (0.01%)
        83μs
        1 x composing: components.forms.tutor-support (0.01%)
        77μs
        1 x composing: components.forms.get-started (0.01%)
        74μs
        1 x composing: components.forms.tutor-subscription-join (0.01%)
        71μs
        1 x composing: components.forms.free-tool-download (0.01%)
        70μs
        1 x eloquent.booted: App\Models\SubjectCat (0.01%)
        65μs
        1 x eloquent.booted: App\Models\Subject (0.01%)
        61μs
        1 x eloquent.booted: App\Models\HomeworkLibrary\HomeworkLibraryFile (0.01%)
        58μs
        1 x composing: site.layouts.app (0%)
        30μs
      14 templates were rendered
      • 1x homework.showshow.blade.phpblade
      • 1x components.breadcrumbsbreadcrumbs.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 college-homework-library/{category}/{subject}/{id}
      middleware
      web, utm.parameters
      controller
      App\Http\Controllers\HomeworkLibraryController@show
      namespace
      where
      as
      homework.show
      file
      app/Http/Controllers/HomeworkLibraryController.php:79-176
      20 statements were executed, 5 of which were duplicates, 15 unique. Show only duplicated78.92ms
      • Connection Establishedtwenty4_siteHomeworkLibraryController.php#91
        Backtrace
        • 13. app/Http/Controllers/HomeworkLibraryController.php:91
        • 14. vendor/laravel/framework/src/Illuminate/Routing/Controller.php:54
        • 15. vendor/laravel/framework/src/Illuminate/Routing/ControllerDispatcher.php:43
        • 16. vendor/laravel/framework/src/Illuminate/Routing/Route.php:260
        • 17. vendor/laravel/framework/src/Illuminate/Routing/Route.php:205
      • select * from `solutionslibrary` where `status` = 'published' and `price` > 0 and `solutionslibrary`.`id` = '38501' limit 1
        9.09mstwenty4_siteHomeworkLibraryController.php#97
        Bindings
        • 0: published
        • 1: 0
        • 2: 38501
        Backtrace
        • 16. app/Http/Controllers/HomeworkLibraryController.php:97
        • 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
        • 20. vendor/laravel/framework/src/Illuminate/Routing/Route.php:205
      • select * from `subjects` where `subjects`.`id` in (71)
        1.28mstwenty4_siteHomeworkLibraryController.php#97
        Backtrace
        • 21. app/Http/Controllers/HomeworkLibraryController.php:97
        • 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
        • 25. vendor/laravel/framework/src/Illuminate/Routing/Route.php:205
      • select * from `solutionslibrary_files` where `solutionslibrary_files`.`solutionlib_id` in (38501)
        5.06mstwenty4_siteHomeworkLibraryController.php#97
        Backtrace
        • 21. app/Http/Controllers/HomeworkLibraryController.php:97
        • 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
        • 25. vendor/laravel/framework/src/Illuminate/Routing/Route.php:205
      • select * from `subject_cats` where `subject_cats`.`id` = 1 limit 1
        1.1mstwenty4_siteHomeworkLibrary.php#201
        Bindings
        • 0: 1
        Backtrace
        • 20. app/Models/HomeworkLibrary/HomeworkLibrary.php:201
        • 26. app/Http/Controllers/HomeworkLibraryController.php:105
        • 27. vendor/laravel/framework/src/Illuminate/Routing/Controller.php:54
        • 28. vendor/laravel/framework/src/Illuminate/Routing/ControllerDispatcher.php:43
        • 29. vendor/laravel/framework/src/Illuminate/Routing/Route.php:260
      • select * from `solutionslibrary` where `id` <> 38501 and `subject` = 71 and `status` = 'published' and `price` > 0 order by RAND() limit 6
        10.38mstwenty4_siteHomeworkLibraryRepository.php#30
        Bindings
        • 0: 38501
        • 1: 71
        • 2: published
        • 3: 0
        Backtrace
        • 14. app/Repositories/HomeworkLibraryRepository.php:30
        • 15. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 16. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 18. app/Repositories/HomeworkLibraryRepository.php:39
        • 19. app/Http/Controllers/HomeworkLibraryController.php:139
      • select * from `subjects` where `subjects`.`id` in (71)
        1.06mstwenty4_siteHomeworkLibraryRepository.php#30
        Backtrace
        • 19. app/Repositories/HomeworkLibraryRepository.php:30
        • 20. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 21. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 23. app/Repositories/HomeworkLibraryRepository.php:39
        • 24. app/Http/Controllers/HomeworkLibraryController.php:139
      • select * from `solutionslibrary_files` where `solutionslibrary_files`.`solutionlib_id` = 38501 and `solutionslibrary_files`.`solutionlib_id` is not null and `publish` = 'question' order by `order` asc, `id` asc
        5.56mstwenty4_siteHomeworkLibrary.php#260
        Bindings
        • 0: 38501
        • 1: question
        Backtrace
        • 15. app/Models/HomeworkLibrary/HomeworkLibrary.php:260
        • 16. app/Transformers/HomeworkLibrary/HomeworkLibraryTransformer.php:58
        • 19. vendor/league/fractal/src/TransformerAbstract.php:128
        • 20. vendor/league/fractal/src/TransformerAbstract.php:107
        • 21. vendor/league/fractal/src/Scope.php:383
      • select * from `solutionslibrary_files` where `solutionslibrary_files`.`solutionlib_id` = 38501 and `solutionslibrary_files`.`solutionlib_id` is not null and `publish` = 'teaser' order by `order` asc, `id` asc
        880μstwenty4_siteHomeworkLibrary.php#260
        Bindings
        • 0: 38501
        • 1: teaser
        Backtrace
        • 15. app/Models/HomeworkLibrary/HomeworkLibrary.php:260
        • 16. app/Transformers/HomeworkLibrary/HomeworkLibraryTransformer.php:69
        • 19. vendor/league/fractal/src/TransformerAbstract.php:128
        • 20. vendor/league/fractal/src/TransformerAbstract.php:107
        • 21. vendor/league/fractal/src/Scope.php:383
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        990μstwenty4_siteHomeworkLibrary.php#260
        Bindings
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        Backtrace
        • 15. app/Models/HomeworkLibrary/HomeworkLibrary.php:260
        • 16. app/Transformers/HomeworkLibrary/HomeworkLibraryTransformer.php:80
        • 19. vendor/league/fractal/src/TransformerAbstract.php:128
        • 20. vendor/league/fractal/src/TransformerAbstract.php:107
        • 21. vendor/league/fractal/src/Scope.php:383
      • select * from `subject_cats` where `subject_cats`.`id` = 1 limit 1
        3.34mstwenty4_siteHomeworkLibrary.php#201
        Bindings
        • 0: 1
        Backtrace
        • 20. app/Models/HomeworkLibrary/HomeworkLibrary.php:201
        • 32. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
        • 33. vendor/laravel/framework/src/Illuminate/View/Engines/PhpEngine.php:58
        • 34. vendor/livewire/livewire/src/ComponentConcerns/RendersLivewireComponents.php:69
        • 35. vendor/laravel/framework/src/Illuminate/View/Engines/CompilerEngine.php:70
      • select * from `subject_cats` where `subject_cats`.`id` = 1 limit 1
        1.25mstwenty4_siteSubject.php#100
        Bindings
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        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` = 3 limit 1
        5.03mstwenty4_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` = 4 limit 1
        2.46mstwenty4_siteSubject.php#100
        Bindings
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        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
        6.59mstwenty4_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 `subject_cats` where `subject_cats`.`id` = 33 limit 1
        2.1mstwenty4_siteSubject.php#100
        Bindings
        • 0: 33
        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` = 11 limit 1
        1.03mstwenty4_siteSubject.php#100
        Bindings
        • 0: 11
        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
        6.43mstwenty4_siteSubject.php#100
        Bindings
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        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
        8.45mstwenty4_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` = 31 limit 1
        900μstwenty4_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 `subject_cats` where `subject_cats`.`id` = 36 limit 1
        5.94mstwenty4_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
        • 28. view::components.footer:170
      App\Models\SubjectCat
      12SubjectCat.php
      App\Models\HomeworkLibrary\HomeworkLibrary
      6HomeworkLibrary.php
      App\Models\HomeworkLibrary\HomeworkLibraryFile
      5HomeworkLibraryFile.php
      App\Models\Subject
      2Subject.php
          _token
          aH76VSte4ZBjKXdRJZGv9HAPOywyDuyF3Rrg5m3L
          utm_source
          direct
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          status_code
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          status_text
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          request_request
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          request_headers
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          request_cookies
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