A PH<math xmlns="http://www.w3.org/1998/Math/MathML" id="M1"> <mfenced open="(" close=")" separators="|"> <mrow> <mi>i</mi> </mrow> </mfenced> </math>/PH<math xmlns="http://www.w3.org/1998/Math/MathML" id="M2"> <mfenced open="(" close=")" separators="|"> <mrow> <mi>i</mi> <mo>,</mo> <mi>n</mi> </mrow> </mfenced> <mo>/</mo> <mrow> <mrow> <mi>C</mi> <mo>/</mo> <mi>C</mi> </mrow> </mrow> </math> Queuing Model in Randomly Changing Environments for Traffic Circulation Systems

Robust optimal design of circulation systems (e.g., roads for vehicles or corridors pedestrians) relies on an accurate steady-state traffic flow model that considers the effect randomly changing environmental factors daily periodicity and weather). Most analytical models assume customer interarrival time service facilities follow exponential distribution with fixed rate parameters, which is unrealistic in most cases. In this paper, we develop a stationary PH i /PH id="M4"> , n / C state-dependent queuing environment (RE), represented by Markov chain. The simultaneously general randomness arrival service, varying (the travel increases number customers). existing matrix analytic scheme (MAS) algorithm extended to solve proposed because it avoids explicit calculation probability distributions. space complexity only linear RE states independent enormous (four-dimensional) state process. Its function product queue capacity states. Our validated versus simulation estimates. obtained conditional performance measures can accurately capture accumulation dissipation reveal environments. Numerical experiments provide some interesting findings. (1) coincides transient M( id="M5"> t )/G id="M6"> x / id="M7"> id="M8"> fluid under special conditions. (2) Under high intensities, increasing duration leads obvious growth length. (3) An increase facility length decrease average output rate, depending whether congestion dissipates effectively one cycle. (4) A larger width required obtain maximum demand greater nonuniformity.