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74-553: The Backbone Trail is a long-distance trail extending 67.79 miles (109.10 km) across the Santa Monica Mountains in the U.S. state of California . Its western terminus is Point Mugu State Park , and its eastern terminus is Will Rogers State Historic Park in Pacific Palisades . The trail is open to hikers throughout its length. Dogs, mountain bicyclists and horseback riders are only allowed on portions of

148-432: A queueing algorithm , which affects the characteristics of the larger network. Mean-field models consider the limiting behaviour of the empirical measure (proportion of queues in different states) as the number of queues m approaches infinity. The impact of other queues on any given queue in the network is approximated by a differential equation. The deterministic model converges to the same stationary distribution as

222-447: A departure rate of μ = avg ( μ 1 , μ 2 , … , μ k ) {\displaystyle \mu ={\text{avg}}(\mu _{1},\mu _{2},\dots ,\mu _{k})} . The steady state equations for the birth-and-death process, known as the balance equations , are as follows. Here P n {\displaystyle P_{n}} denotes

296-785: A huge split boulder known as Split Rock. The return route on the Backbone Trail leads to Inspiration Point and culminates at Sandstone Peak, the highest point in the Santa Monica Mountains. Both points overlook the Pacific Ocean, the Channel Islands, and the surrounding mountains. Parts of the trail lie within the city limits of Los Angeles and Malibu, California ; the rest sits in unincorporated areas of Los Angeles County and Ventura County . The following national, state, and privately administered wilderness areas are traversed by

370-899: A large number of trekkers ( backpackers ). Typical trekking regions in Nepal are Annapurna , Dolpo , Langtang , Manaslu , Kangchenjunga and Mount Everest . In India, the Kashmir Valley is home to several trekking routes that traverse western sections of the Himalayas. Vishansar Lake , Gangabal Lake and Tarsar Lakes are accessible only through different trekking routes. Other popular trekking routes in India include Chandra Taal , Dzongri, Goechala, Gomukh , Hemkund , Kafni Glacier , Kailash - Manasarovar , Kedarnath , Kedartal , Milam Glacier , Nanda Devi Sanctuary , Pindari Glacier , Richenpong , Roopkund , Sar Pass , Satopanth Tal , Saurkundi Pass and

444-410: A queue with no buffer (or no waiting area ). A setting with a waiting zone for up to n customers is called a queue with a buffer of size n . The behaviour of a single queue (also called a queueing node ) can be described by a birth–death process , which describes the arrivals and departures from the queue, along with the number of jobs currently in the system. If k denotes the number of jobs in

518-487: A queue, these rates are generally considered not to vary with the number of jobs in the queue, so a single average rate of arrivals/departures per unit time is assumed. Under this assumption, this process has an arrival rate of λ = avg ( λ 1 , λ 2 , … , λ k ) {\displaystyle \lambda ={\text{avg}}(\lambda _{1},\lambda _{2},\dots ,\lambda _{k})} and

592-631: A queue. Queue networks are systems in which multiple queues are connected by customer routing . When a customer is serviced at one node, it can join another node and queue for service, or leave the network. For networks of m nodes, the state of the system can be described by an m –dimensional vector ( x 1 , x 2 , ..., x m ) where x i represents the number of customers at each node. The simplest non-trivial networks of queues are called tandem queues . The first significant results in this area were Jackson networks , for which an efficient product-form stationary distribution exists and

666-418: A stochastic (random) process (usually Poisson) and are followed by setup periods during which the server is unavailable. The interrupted customer remains in the service area until server is fixed. Arriving customers not served (either due to the queue having no buffer, or due to balking or reneging by the customer) are also known as dropouts . The average rate of dropouts is a significant parameter describing

740-410: A well-maintained, single-track path with some dirt fire roads along the route. The trail is very well designed with climbs and descents occurring on well-graded trails full of switchbacks. The trail can be hiked in either the westbound or eastbound direction. There are no major differences between the two directions. Hiking westbound, you will progress towards more remote territory and can end your hike at

814-480: Is a longer recreational trail mainly through rural areas used for hiking , backpacking , cycling , horse riding or cross-country skiing . They exist on all continents except Antarctica. Many trails are marked on maps. Typically, a long-distance route will be at least 50 km (30 mi) long, but many run for several hundred miles, or longer. Many routes are waymarked and may cross public or private land and/or follow existing rights of way . Generally,

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888-543: Is applicable to other rail trails that exist throughout the world: Following the route of the railways, they cut through hills, under roads, over embankments and across gullies and creeks. Apart from being great places to walk, cycle or horse ride, rail trails are linear conservation corridors protecting native plants and animals. They often link remnant vegetation in farming areas and contain valuable flora and fauna habitat. Wineries and other attractions are near many trails as well as B&B's and other great places to stay. In

962-416: Is essential in contexts such as traffic systems, computer networks, telecommunications, and service operations. Queueing theory delves into various foundational concepts, with the arrival process and service process being central. The arrival process describes the manner in which entities join the queue over time, often modeled using stochastic processes like Poisson processes. The efficiency of queueing systems

1036-596: Is expected to open by the end of 2024 as the longest coastal walking route in the world and Britain's longest National Trail . These may be cross-country paths, or may follow roads or other ways, and often intersect with many other trails. Examples are Wainwright's Coast-to-Coast path in northern England, and the GR 10 in France . The English Coast to Coast route, despite being amongst the best-known long-distance walking routes in England,

1110-529: Is not an official National Trail , but simply a series of connected pre-existing rights of way, roads and open country with some informal links between them. There is also a coast-to-coast mountain-bike route in northern England that has the same trailheads as the walkers' path. GR 10 is a French GR footpath that runs the length of the Pyrenees Mountains , roughly paralleling the French–Spanish border on

1184-711: Is published in April in the association's magazine, Strider. The Kerry Way , in south-west Ireland, is the longest of the Irish waymarked trails and circumnavigates the highest mountain range in Ireland. Along with the adjoining Dingle Way it is noted for its scenic views of the Atlantic , loughs and mountains. Long-distance trails in Hong Kong : Japan has a network of ten long-distance trails called Long Distance Nature Trails. Their creation

1258-537: Is shown to also exhibit a product–form stationary distribution. The normalizing constant can be calculated with the Buzen's algorithm , proposed in 1973. Networks of customers have also been investigated, such as Kelly networks , where customers of different classes experience different priority levels at different service nodes. Another type of network are G-networks , first proposed by Erol Gelenbe in 1993: these networks do not assume exponential time distributions like

1332-403: Is that of the cashier at a supermarket. (There are other models, but this one is commonly encountered in the literature.) Customers arrive, are processed by the cashier, and depart. Each cashier processes one customer at a time, and hence this is a queueing node with only one server. A setting where a customer will leave immediately if the cashier is busy when the customer arrives, is referred to as

1406-455: Is the longest marked multi-use trail in the world, stretching 5,330 kilometres (3,310 mi) from Cooktown, Queensland , through New South Wales to Healesville, Victoria . This non-motorised trail runs the length of the rugged Great Dividing Range through national parks and private property and alongside wilderness areas . One of the objectives was to develop a trail that linked up the brumby tracks, mustering and stock routes along

1480-600: Is the result of a Ministry of Environment initiative to highlight the specific environmental, cultural or historic landscapes through which the trails pass. They also aim to allow hikers a safe and easy hike in any season, as well as raising awareness of the importance of protecting natural spaces and adopting sustainable behaviors. Long Distance Nature Trails in Japan : In Brazil, long-distance trails are regulated by two Federal Government decrees, and implemented and managed by government agencies in partnership with many NGOs, such as

1554-515: Is to compute these characteristics for the current system and then test several alternatives that could lead to improvement. Computing the operating characteristics for the current system and comparing the values to the characteristics of the alternative systems allows managers to see the pros and cons of each potential option. These systems help in the final decision making process by showing ways to increase savings, reduce waiting time, improve efficiency, etc. The main queueing models that can be used are

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1628-743: The EuroVelo routes. Some trails follow the towpaths of canal systems. A good example is the 845-kilometre (525 mi) New York State Canal System in New York . There also numerous routes that can be followed in Europe, which may be suitable for walkers, cyclists, horse riders and canoeists. Many long-distance trails have sections suitable for equestrians, and a few are suitable for horse riding throughout their length, or have been developed primarily for horse riding. The Bicentennial National Trail (BNT) in Australia

1702-704: The Iron Curtain Trail (also known as EuroVelo 13). The latter is a partially complete long-distance cycling route which will run along the entire length of the former Iron Curtain . During the period of the Cold War (c. 1947–1991), the Iron Curtain delineated the border between the Communist East and the capitalist West. Some of the longest walking routes worldwide: Long-distance mountain trails are of two broad kinds: linear trails and loop trails. In Europe

1776-650: The Pennine Bridleway , 192 km (119 miles), The Ridgeway , 139 km (86 miles), and the South Downs Way , 160 km (99 miles). Rail trails (or rail paths) are shared-use paths that make use of abandoned railway corridors. There are also rails with trails in the US that follow working rail tracks. Most rail trails have a gravel or dirt surface and can be used for walking, cycling, and often horse riding as well. The following description comes from Australia , but

1850-639: The Pollaczek–Khinchine formula . After the 1940s, queueing theory became an area of research interest to mathematicians. In 1953, David George Kendall solved the GI/M/ k queue and introduced the modern notation for queues, now known as Kendall's notation . In 1957, Pollaczek studied the GI/G/1 using an integral equation . John Kingman gave a formula for the mean waiting time in a G/G/1 queue , now known as Kingman's formula . Leonard Kleinrock worked on

1924-829: The South West Coast Path . The equivalent routes in Scotland are styled as Scotland's Great Trails ; they include the West Highland Way and the Speyside Way . The success of the Welsh government's 870-mile Wales Coast Path prompted an ongoing project of create a similar route for England. When completed, the King Charles III England Coast Path will be around 2,700 miles long. There are many other recognised, sometimes waymarked, long-distance footpaths in

1998-612: The Valley of Flowers . The Great Himalaya Trail is proposed to follow the Greater Himalaya Range from Namche Barwa in Tibet to Nanga Parbat in Pakistan, forming the world's highest mountain trail. A long-distance trail network in the southern Andes, the 3,000-kilometre (1,900 mi) Greater Patagonian Trail , was first described in 2014. It currently connects Santiago de Chile with

2072-644: The Via Alpina consists of five connected hiking trails across the alpine regions of Slovenia , Austria , Germany , Liechtenstein , Switzerland , Italy , France and Monaco . It is 5,000 km (3,100 miles) long, with 342 day stages. Circular routes include the Tour du Mont Blanc , which passes through the Alps of France, Switzerland, and Italy. In the Balkans region, the Peaks of

2146-440: The geometric distribution formula where ρ = λ μ < 1 {\displaystyle \rho ={\frac {\lambda }{\mu }}<1} . A common basic queueing system is attributed to Erlang and is a modification of Little's Law . Given an arrival rate λ , a dropout rate σ , and a departure rate μ , length of the queue L is defined as: Assuming an exponential distribution for

2220-551: The mean value analysis (which allows average metrics such as throughput and sojourn times) can be computed. If the total number of customers in the network remains constant, the network is called a closed network and has been shown to also have a product–form stationary distribution by the Gordon–Newell theorem . This result was extended to the BCMP network , where a network with very general service time, regimes, and customer routing

2294-964: The 27 Brazilian states , connecting all Brazilian biomes. As of January 2022, Brazil has more than 5,500 km of managed trails and another 20,500 km planned. Long-distance trails in Brazil: These follow coastlines; examples are the Brittany Coast Path in France, the California Coastal Trail in the US, the South West Coast Path in England, the East Coast Trail in Canada, and the Otter Trail in South Africa. The King Charles III England Coast Path , in development by Natural England , will be around 4,350 kilometres (2,700 mi) long. It

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2368-468: The Atlantic Ocean and the western terminus is Point Reyes , on the northern California coast at the Pacific Ocean. The Iditarod Trail connects the coastal cities of Seward and Nome, Alaska : a distance of around 1,600 kilometres (990 mi). The European long-distance paths (E-paths) traverse Europe, passing through many different countries. Among the longest are European walking route E8 and

2442-568: The BBT, and they are not evenly spaced out, posing a logistical challenge for an end-to-end hike. Backpacking the full length of the BBT with reasonable daily mileages may require the use of off-trail camping and lodging options, which are available within a reasonable distance from the trail. 34°07′10″N 118°56′15″W  /  34.11944°N 118.93750°W  / 34.11944; -118.93750 Long distance trail A long-distance trail (or long-distance footpath , track, way, greenway )

2516-471: The Backbone Trail was designated a National Recreation Trail . The official opening ceremony for the Backbone Trail was held on Saturday, June 4, 2016, at Will Rogers State Park, with a ribbon-cutting ceremony held at 10:00 am. Portions of the trail were closed after the Woolsey Fire in 2018. Other trails and several parks were closed as 88% of the federal parkland within the Santa Monica Mountains burned in

2590-466: The Backbone Trail: Chumash villages once stood throughout the Santa Monica Mountains. The California chaparral and woodlands and California coastal sage and chaparral ecoregions supported Indigenous people for tens of thousands of years. A wide range of flora and fauna can be observed from the trail. For the majority of its length, the Backbone Trail passes through terrain occupied by

2664-769: The Balkans Trail and High Scardus Trail connect Albania , Kosovo and Montenegro or North Macedonia respectively through a network of combined almost 700 kilometres (430 mi). In the United States, notable linear trails include the Appalachian Trail , 3,500 kilometres (2,200 mi), the Pacific Crest Trail , 4,300 kilometres (2,700 mi) and the Continental Divide Trail , 5,000 kilometres (3,100 mi). The first long-distance hiking trail in

2738-631: The Brazilian Trails Network Association ( Associação Rede Brasileira de Trilhas in Portuguese) and the Atlantic Forest Trail Institute. The aim is to create a national system of trails that are pleasant to hike, but that also generate employment and income and function as conservation tools by linking protected areas with natural corridors. There are more than 120 trails in different stages of implementation in 25 of

2812-639: The French side. It runs west to east, from Hendaye on the Bay of Biscay to Banyuls-sur-Mer on the Mediterranean Sea . The American Discovery Trail is a hiking and biking trail that crosses the continental United States from east to west, across the mid-tier of the United States 10,900 kilometres (6,800 mi). Horses can also be ridden on most of this trail. The eastern terminus is the Delmarva Peninsula on

2886-403: The G stands for "general" and indicates an arbitrary probability distribution for service times. Consider a queue with one server and the following characteristics: Further, let E n {\displaystyle E_{n}} represent the number of times the system enters state n , and L n {\displaystyle L_{n}} represent the number of times

2960-767: The Great Dividing Range, thus allowing one legally to ride the routes of stockmen and drovers who once traveled these areas with pack horses . The Bicentennial National Trail is suitable for self-reliant horse riders, fit walkers and mountain bike riders. In the United Kingdom, the British Horse Society is developing a network of horse trails known as the National Bridleroute Network . A number of long-distance multi-use trails have been created in England, including three National Trails :

3034-563: The Netherlands, Spain and Portugal (the Alta Via (Italian), Grande Randonnée (French), Grote Routepaden or Lange-afstand-wandelpaden (Dutch), Grande Rota (Portuguese) or Gran Recorrido (Spanish)). National Trails are a network of officially sanctioned footpaths in the United Kingdom which are well maintained and well waymarked across England and Wales. Examples are the Pennine Way and

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3108-484: The Pacific Ocean, but transportation may be more difficult from the Ray Miller Trailhead. There is no permit required for hiking the BBT. Camping along the Backbone Trail is restricted to established campgrounds. These include group campsites that can be reserved in advance and walk-in campsites that are available on a first-come, first-served basis. There are still relatively few established campgrounds right on

3182-480: The Santa Monica Mountains ), the trail is more frequently graded than level. Scenic views of Ventura County are seen from the six-mile loop trail, starting at the trailhead off Yerba Buena road. The Mishe Mokwa Trail in Circle X Ranch follows Carlisle Canyon along Boney Mountain past weathered red volcanic formations. There are views of the sculpted caves and crevices of Echo Cliffs and a forested streamside picnic area by

3256-785: The Southern Patagonian Icefield and explores the remote areas of the Patagonian Andes in the border region between Chile and Argentina. The entire network currently incorporates more than 16,000-kilometre (9,900 mi) of routes and provides many packrafting options. These routes have been constructed mainly for bicycle touring . Some are restricted to use by only non-motorized bikes while others are multi-use recreational (i.e. hiking, horseback riding, jogging, rollerblading or walking). Some long-distance cycling routes are hundreds of miles long, such as Australia's mainly off-road Munda Biddi Trail , or even thousands of miles, such as

3330-615: The UK which do not have National Trail status. The Long Distance Walkers Association (LDWA) has the most comprehensive online database of long-distance paths in the UK, and members are able to download GPX files of routes. The association also maintains the LDWA National Trails Register, with different levels of membership for people who have completed five, 10, 15 or all 19 of the National Trails and Great Trails. An annual report

3404-693: The US was begun in 1910 and named The Long Trail . Notable circular trails include the Tahoe Rim Trail and the Wonderland Trail (which encircles Mount Rainier ). The Australian Alps Walking Track traverses the alpine areas of Victoria , New South Wales and the Australian Capital Territory . It is 655 km (407 miles) long, starting at Walhalla, Victoria , and running through to Tharwa, Australian Capital Territory near Canberra . The Himalayan routes are famous for attracting

3478-523: The US, the 43-kilometre (27 mi) Cheshire Rail Trail , in New Hampshire , can be used by hikers, horseback riders, snowmobilers, cross-country skiers, cyclists, or even dog-sledders. In Canada , following the abandonment of the Prince Edward Island Railway in 1989, the government of Prince Edward Island purchased the right-of-way to the entire railway system. The Confederation Trail

3552-823: The application of queueing theory to message switching in the early 1960s and packet switching in the early 1970s. His initial contribution to this field was his doctoral thesis at the Massachusetts Institute of Technology in 1962, published in book form in 1964. His theoretical work published in the early 1970s underpinned the use of packet switching in the ARPANET , a forerunner to the Internet. The matrix geometric method and matrix analytic methods have allowed queues with phase-type distributed inter-arrival and service time distributions to be considered. Systems with coupled orbits are an important part in queueing theory in

3626-520: The application to wireless networks and signal processing. Modern day application of queueing theory concerns among other things product development where (material) products have a spatiotemporal existence, in the sense that products have a certain volume and a certain duration. Problems such as performance metrics for the M/G/ k queue remain an open problem. Various scheduling policies can be used at queueing nodes: Server failures occur according to

3700-440: The classic Jackson network. In discrete-time networks where there is a constraint on which service nodes can be active at any time, the max-weight scheduling algorithm chooses a service policy to give optimal throughput in the case that each job visits only a single-person service node. In the more general case where jobs can visit more than one node, backpressure routing gives optimal throughput. A network scheduler must choose

3774-452: The distribution of service times for jobs, and c the number of servers at the node. For an example of the notation, the M/M/1 queue is a simple model where a single server serves jobs that arrive according to a Poisson process (where inter-arrival durations are exponentially distributed ) and have exponentially distributed service times (the M denotes a Markov process ). In an M/G/1 queue ,

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3848-398: The equation for P n {\displaystyle P_{n}} ( n ≥ 1 ) {\displaystyle (n\geq 1)} , fully describes the required steady state probabilities. Single queueing nodes are usually described using Kendall's notation in the form A/S/ c where A describes the distribution of durations between each arrival to the queue, S

3922-413: The fast-moving fire. The Backbone Trail fully reopened in July 2019. As the Backbone Trail passes over the peaks and through the canyons of the Santa Monica Mountains, it climbs and falls in elevation nearly constantly. From its lowest point, the sea-level Ray Miller Trailhead at Point Mugu State Park, to its highest point, near Sandstone Peak (which, at 3,111 ft (948 m), is the highest peak in

3996-490: The field is Queueing Systems . Queueing theory is one of the major areas of study in the discipline of management science . Through management science, businesses are able to solve a variety of problems using different scientific and mathematical approaches. Queueing analysis is the probabilistic analysis of waiting lines, and thus the results, also referred to as the operating characteristics, are probabilistic rather than deterministic. The probability that n customers are in

4070-421: The field of teletraffic engineering and have since seen applications in telecommunications , traffic engineering , computing , project management , and particularly industrial engineering , where they are applied in the design of factories, shops, offices, and hospitals. The spelling "queueing" over "queuing" is typically encountered in the academic research field. In fact, one of the flagship journals of

4144-468: The field) arrive to the queue, possibly wait some time, take some time being processed, and then depart from the queue. However, the queueing node is not quite a pure black box since some information is needed about the inside of the queueing node. The queue has one or more servers which can each be paired with an arriving job. When the job is completed and departs, that server will again be free to be paired with another arriving job. An analogy often used

4218-485: The first paper on what would now be called queueing theory. He modeled the number of telephone calls arriving at an exchange by a Poisson process and solved the M/D/1 queue in 1917 and M/D/ k queueing model in 1920. In Kendall's notation: If the node has more jobs than servers, then jobs will queue and wait for service. The M/G/1 queue was solved by Felix Pollaczek in 1930, a solution later recast in probabilistic terms by Aleksandr Khinchin and now known as

4292-560: The future ( E n = L n {\displaystyle E_{n}=L_{n}} ) or not ( | E n − L n | = 1 {\displaystyle \left\vert E_{n}-L_{n}\right\vert =1} ). When the system arrives at a steady state, the arrival rate should be equal to the departure rate. Thus the balance equations imply The fact that P 0 + P 1 + ⋯ = 1 {\displaystyle P_{0}+P_{1}+\cdots =1} leads to

4366-475: The ground-hugging bushes and shrubs that characterize the coastal sage scrub and chaparral ecosystems. Relatively few trees line the trail, an essential consideration for travelers looking for shade. More than 450 vertebrate species have been observed within the Santa Monica Mountains NRA, including more than 380 species of birds. Lizards, seven species of which inhabit areas surrounding the trail, are particularly numerous, and can be easily found on or alongside

4440-651: The limit when the process is scaled in time and space, allowing heterogeneous objects. This scaled trajectory converges to a deterministic equation which allows the stability of the system to be proven. It is known that a queueing network can be stable but have an unstable fluid limit. Queueing theory finds widespread application in computer science and information technology. In networking, for instance, queues are integral to routers and switches, where packets queue up for transmission. By applying queueing theory principles, designers can optimize these systems, ensuring responsive performance and efficient resource utilization. Beyond

4514-417: The most effective method. Queueing theory, a discipline rooted in applied mathematics and computer science, is a field dedicated to the study and analysis of queues, or waiting lines, and their implications across a diverse range of applications. This theoretical framework has proven instrumental in understanding and optimizing the efficiency of systems characterized by the presence of queues. The study of queues

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4588-514: The original model. In a system with high occupancy rates (utilisation near 1), a heavy traffic approximation can be used to approximate the queueing length process by a reflected Brownian motion , Ornstein–Uhlenbeck process , or more general diffusion process . The number of dimensions of the Brownian process is equal to the number of queueing nodes, with the diffusion restricted to the non-negative orthant . Fluid models are continuous deterministic analogs of queueing networks obtained by taking

4662-439: The queueing system, the average number of customers in the queueing system, the average number of customers in the waiting line, the average time spent by a customer in the total queuing system, the average time spent by a customer in the waiting line, and finally the probability that the server is busy or idle are all of the different operating characteristics that these queueing models compute. The overall goal of queueing analysis

4736-425: The rates, the waiting time W can be defined as the proportion of arrivals that are served. This is equal to the exponential survival rate of those who do not drop out over the waiting period, giving: The second equation is commonly rewritten as: The two-stage one-box model is common in epidemiology . In 1909, Agner Krarup Erlang , a Danish engineer who worked for the Copenhagen Telephone Exchange, published

4810-424: The single-server waiting line system and the multiple-server waiting line system, which are discussed further below. These models can be further differentiated depending on whether service times are constant or undefined, the queue length is finite, the calling population is finite, etc. A queue or queueing node can be thought of as nearly a black box . Jobs (also called customers or requests , depending on

4884-629: The steady state probability to be in state n . The first two equations imply and By mathematical induction, The condition ∑ n = 0 ∞ P n = P 0 + P 0 ∑ n = 1 ∞ ∏ i = 0 n − 1 λ i μ i + 1 = 1 {\displaystyle \sum _{n=0}^{\infty }P_{n}=P_{0}+P_{0}\sum _{n=1}^{\infty }\prod _{i=0}^{n-1}{\frac {\lambda _{i}}{\mu _{i+1}}}=1} leads to which, together with

4958-644: The surface is not specially prepared, and the ground can be rough and uneven in areas, except in places such as converted rail tracks or popular walking routes where stone-pitching and slabs have been laid to prevent erosion. In some places, official trails will have the surface specially prepared to make the going easier. Historically, and still nowadays in countries where most people move on foot or with pack animals, long-distance trails linked far away towns and regions. Such paths followed "logical" routes, that can be approximated to least-cost paths . GR footpaths are long-distance footpaths in Italy, France, Belgium,

5032-505: The system (either being serviced or waiting if the queue has a buffer of waiting jobs), then an arrival increases k by 1 and a departure decreases k by 1. The system transitions between values of k by "births" and "deaths", which occur at the arrival rates λ i {\displaystyle \lambda _{i}} and the departure rates μ i {\displaystyle \mu _{i}} for each job i {\displaystyle i} . For

5106-419: The system leaves state n . Then | E n − L n | ∈ { 0 , 1 } {\displaystyle \left\vert E_{n}-L_{n}\right\vert \in \{0,1\}} for all n . That is, the number of times the system leaves a state differs by at most 1 from the number of times it enters that state, since it will either return into that state at some time in

5180-411: The technological realm, queueing theory is relevant to everyday experiences. Whether waiting in line at a supermarket or for public transportation, understanding the principles of queueing theory provides valuable insights into optimizing these systems for enhanced user satisfaction. At some point, everyone will be involved in an aspect of queuing. What some may view to be an inconvenience could possibly be

5254-469: The trail as posted. Construction on the backbone trail began in the early 1980s. It involved the cooperation of the National Park Service , the California Department of Parks and Recreation , and private parties interested in connecting existing trails within the Santa Monica Mountains NRA into a single trail spanning the length of the park. The range of conditions found on the trail - ranging from flat, wide fire roads to narrow single-track trails - reflects

5328-587: The trail's origin as a series of unconnected paths and backcountry roads, connected to form the Backbone Trail System. In 2016, the National Park Service completed land acquisitions from former California Governor Arnold Schwarzenegger and fitness pioneer Betty Weider to enable right-of-way for the trail. In May 2016, a new section of the trail was completed on the east side of Yerba Buena Road in support of these acquisitions. Upon its 2016 opening,

5402-527: The trail. Southern Pacific rattlesnakes can be numerous in early spring, particularly along Temescal Ridge in the eastern section of the trail. Mountain lions have been observed in the western section near Sandstone Peak trailhead in June 2008. At least eight mountain lions are believed to remain from Pt. Mugu to Topanga State Park. Notable species that can found along the Backbone trail include: The Backbone Trail follows

5476-585: Was developed as a tip-to-tip walking and cycling gravel rail trail which doubles as a monitored and groomed snowmobile trail during the winter months, operated by the PEI Snowmobile Association. First-come, first-served Queueing theory has its origins in research by Agner Krarup Erlang , who created models to describe the system of incoming calls at the Copenhagen Telephone Exchange Company. These ideas were seminal to

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