[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-seo-657288-105":59,"doc-detail-657288-en":130},{"code":4,"msg":5,"data":6},0,"success",[7,13,18,23,28,33,38,43,48,51,55],{"id":8,"doc_module":4,"doc_module_name":9,"category_name":10,"show_sort_weight":11,"slug":12},1,"Document","Story & Novel",90,"story-novel",{"id":14,"doc_module":4,"doc_module_name":9,"category_name":15,"show_sort_weight":16,"slug":17},2,"Literature",80,"literature",{"id":19,"doc_module":4,"doc_module_name":9,"category_name":20,"show_sort_weight":21,"slug":22},4,"Exam",70,"exam",{"id":24,"doc_module":4,"doc_module_name":9,"category_name":25,"show_sort_weight":26,"slug":27},5,"Comic",60,"comic",{"id":29,"doc_module":4,"doc_module_name":9,"category_name":30,"show_sort_weight":31,"slug":32},6,"Technology",50,"technology",{"id":34,"doc_module":4,"doc_module_name":9,"category_name":35,"show_sort_weight":36,"slug":37},7,"Healthcare",40,"healthcare",{"id":39,"doc_module":4,"doc_module_name":9,"category_name":40,"show_sort_weight":41,"slug":42},8,"Research & Report",30,"research-report",{"id":44,"doc_module":4,"doc_module_name":9,"category_name":45,"show_sort_weight":46,"slug":47},9,"Religion & Spirituality",20,"religion-spirituality",{"id":46,"doc_module":4,"doc_module_name":9,"category_name":49,"show_sort_weight":46,"slug":50},"World Cup","world-cup",{"id":52,"doc_module":4,"doc_module_name":9,"category_name":53,"show_sort_weight":52,"slug":54},10,"Lifestyle","lifestyle",{"id":56,"doc_module":4,"doc_module_name":9,"category_name":57,"show_sort_weight":24,"slug":58},19,"General","general",{"code":4,"msg":60,"data":61},"ok",{"site_id":62,"language":63,"slug":64,"title":65,"keywords":66,"description":67,"schema_data":68,"social_meta":123,"head_meta":125,"extra_data":127,"updated_unix":129},105,"en","cumulative-diagrams-an-example","Cumulative Diagrams - An Example","","Explores airport runway queuing under a temporary capacity reduction using a deterministic model. The demand rate O(t) is constant during the busy period, while capacity drops from normal high h to low l between times a and b, with T=b−a. Derives the queue growth rate, the total time with a queue via a triangular area, and the total delay time incurred by delayed aircraft. Shows that expected delay per delayed aircraft depends only on low capacity and demand rate, not on the high capacity. Introduces cumulative flow diagrams as the graphical tool for demands versus admissions.",{"@graph":69,"@context":122},[70,84,105],{"@type":71,"itemListElement":72},"BreadcrumbList",[73,77,79,82],{"item":74,"name":75,"@type":76,"position":8},"https://docshare.wps.com","Home","ListItem",{"item":78,"name":9,"@type":76,"position":14},"https://docshare.wps.com/document/",{"item":80,"name":40,"@type":76,"position":81},"https://docshare.wps.com/document/research-report/",3,{"item":83,"name":65,"@type":76,"position":19},"https://docshare.wps.com/document/cumulative-diagrams-an-example/657288/",{"url":83,"name":65,"@type":85,"image":86,"author":91,"headline":65,"publisher":94,"fileFormat":97,"inLanguage":63,"description":67,"dateModified":98,"datePublished":99,"encodingFormat":97,"isAccessibleForFree":100,"interactionStatistic":101},"DigitalDocument",{"url":87,"@type":88,"width":89,"height":90},"https://docshare.wps.com/thumbnails/cumulative-diagrams-an-example/657288.png","ImageObject",300,407,{"name":92,"@type":93},"gewufjwa","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-10-29","2026-10-26",true,{"@type":102,"interactionType":103,"userInteractionCount":14},"InteractionCounter",{"@type":104},"ViewAction",{"@type":106,"mainEntity":107},"FAQPage",[108,114,118],{"name":109,"@type":110,"acceptedAnswer":111},"What do O(t) and P(t) represent in the model?","Question",{"text":112,"@type":113},"O(t) denotes the demand rate (expected requests per unit time) and P(t) denotes the service rate or capacity (maximum throughput when the runway is continually busy).","Answer",{"name":115,"@type":110,"acceptedAnswer":116},"How is the duration of low capacity defined?",{"text":117,"@type":113},"Low capacity lasts from time t=a to t=b, and the interval length is defined as T=b−a.",{"name":119,"@type":110,"acceptedAnswer":120},"Why does the expected delay per delayed aircraft not depend on the high capacity h?",{"text":121,"@type":113},"The derivation shows that the expected delay per delayed aircraft is a function only of the low capacity l and the demand rate n, while the result is independent of the high capacity h.","https://schema.org",{"og:url":83,"og:type":124,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":126,"canonical":83},"index,follow",{"doc_id":128,"site_id":62},657288,1792977039,{"code":4,"msg":5,"data":131},{"doc_id":128,"user_id":132,"nickname":92,"user_avatar":133,"doc_module":4,"category_id":39,"category_name":40,"doc_title":65,"doc_description":67,"doc_content":134,"file_id":135,"file_url":136,"file_type":137,"file_size":138,"view_count":14,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":34,"language":139,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":140,"faqs":141,"seo_title":142,"seo_description":67,"update_tm":129,"read_time":143},3985747882900,"https://ap-avatar.wpscdn.com/davatar_9964176cb1d06d4a9deccf72a44ae3dc","Cumulative Diagrams: An Example  \nConsider Figure 1 in which the functions O(t) and P(t) denote, respectively, the demand rate and the service rate (or “capacity”) over time at the runway system of an airport 1. The demand rate is the expected number of demands per unit of time, i.e., the number of requests for service at the runway system per unit of time. Note that not all these requests will be necessarily satisfied due to capacity limitations. Similarly the service rate is the expected number of service requests that can be satisfied per unit of time when the runway system is continually busy, i.e., it is the maximum throughput capacity of the runway system (details in the next lecture of our course) . The units we shall use for both the demand rate and the capacity are “aircraft movements (arrivals or departures) per hour”, unless otherwise stated.  \nFigure 1 approximates a situation that one sees all the time at busy airports. The time axis shows the busy part ofa day at the airport, e.g., the origin may correspond to 07:00 local time. To simplify the analysis we assume the demand rate is constant throughout the busy part of the day, i.e., O(t) = n movements per hour for all t. While the normal capacity h (for “high”) is greater than n, a weather event occurs between t = a and t = b (e.g., this could be a period when fog is present during a particular day) which reduces the capacity to l (for “low”) movements per hour. It will be convenient later onto denote with T = b – a the interval of time during which the capacity is low. We would like to explore quantitatively the implications of this temporary loss of capacity on delay levels at the airport.  \n1 We shall use throughout the Greek symbols Oand P to denote the expected “demand rate” and expected “service rate”(or “capacity”); this is consistent with the standard notation used in queuing theory.  \nWe shall make another important simplification in our model at this point. Namely, we shall assume that demands occur at evenly spaced intervals (e.g., if n = 60, a demand occurs exactly every 60 seconds) and, similarly, for any value of the service rate, h or l, the service times are constant (e.g., if l = 30 and the runway system is continually busy, an aircraft movement is completed exactly every 2 minutes) . In queuing theory, this is called a deterministic model, by contrast to probabilistic models in which either the times between the occurrence of successive demands or the service times (or, more usually, both) are random variables, i.e., may take on different values, each value associated with a certain probability.  \nClearly the situation shown in Figure 1 will result in some delays to aircraft movements during the time period between t = a and t = b, at the very least. In fact, it is very easy to plot, as a function of time, the number of aircraft (landing or taking off) which will be in the queue, waiting to use the runway system. This is done in Figure 2. There is no queue until t = a because the runway system’s capacity is greater than the demand rate. Beginning at t = a the queue builds up at a rate ofn – l aircraft per unit of time, so that, by the time b, the queue will build up to a length of (n-l)􀂘(b-a) = (n-l)􀂘T aircraft. After time b, when the runway system’s capacity is “high” again, the queue length will decrease at the rate of h-n aircraft per unit of time, i.e., at the rate at which aircraft receive service minus the rate at which new aircraft join the queue. Thus, it will  \ntake an amount of time equal to (n 􀀐 l) 􀂘  \nT (h  \n􀀐 n)  \nfor the queue to dissipate and get back  \nto zero aircraft. We therefore have:  \n(n 􀀐 l) 􀂘T (h 􀀐 l)   \nTotal amount of time with a queue present = T 􀀎 T 􀂘 (1)  \n(h 􀀐 n) (h 􀀐 n)  \nThe area under the triangle in Figure 2 represents the total amount of time that aircraft will spend waiting in the queue, i.e., the total delay time suffered by all delayed aircraft due to the reduction in the capacity from t = a to t = b. We have","cbCaiqON2jnHNbFO","https://ap.wps.com/l/cbCaiqON2jnHNbFO","pdf",62166,"English","# Cumulative Diagrams: An Example\n## Demand and Service Rates Over Time\n## Deterministic Assumptions\n## Queue Growth and Dissipation\n## Total Delay Time and Expected Delay\n## Cumulative Flow Diagrams (Figures 3)","[{\"question\":\"What do O(t) and P(t) represent in the model?\",\"answer\":\"O(t) denotes the demand rate (expected requests per unit time) and P(t) denotes the service rate or capacity (maximum throughput when the runway is continually busy).\"},{\"question\":\"How is the duration of low capacity defined?\",\"answer\":\"Low capacity lasts from time t=a to t=b, and the interval length is defined as T=b−a.\"},{\"question\":\"Why does the expected delay per delayed aircraft not depend on the high capacity h?\",\"answer\":\"The derivation shows that the expected delay per delayed aircraft is a function only of the low capacity l and the demand rate n, while the result is independent of the high capacity h.\"}]","Cumulative Diagrams - An Example | PDF",18]