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

Robust optimal design of circulation systems (e.g., roads for vehicles or corridors for pedestrians) relies on an accurate steady-state traffic flow model that considers the effect of randomly changing environmental factors (e.g., daily periodicity and weather). Most analytical models assume that the customer interarrival time and service time of circulation facilities follow the exponential distribution with fixed rate parameters, which is unrealistic in most cases. In this paper, we develop a stationary PH <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" id="M3"> <a:mfenced open="(" close=")" separators="|"> <a:mrow> <a:mi>i</a:mi> </a:mrow> </a:mfenced> </a:math> /PH <f:math xmlns:f="http://www.w3.org/1998/Math/MathML" id="M4"> <f:mfenced open="(" close=")" separators="|"> <f:mrow> <f:mrow> <f:mrow> <f:mfenced open="(" close=")" separators="|"> <f:mrow> <f:mi>i</f:mi> <f:mo>,</f:mo> <f:mi>n</f:mi> </f:mrow> </f:mfenced> </f:mrow> <f:mo>/</f:mo> <f:mrow> <f:mfenced open="(" close=")" separators="|"> <f:mrow> <f:mrow> <f:mi>C</f:mi> <f:mo>/</f:mo> <f:mi>C</f:mi> </f:mrow> </f:mrow> </f:mfenced> </f:mrow> </f:mrow> </f:mrow> </f:mfenced> </f:math> state-dependent queuing model in a randomly changing environment (RE), which is represented by a Markov chain. The model simultaneously considers the general randomness of arrival and service, the randomly varying rate parameters, and the state-dependent service (the travel time increases with the number of customers). The existing matrix analytic scheme (MAS) algorithm is extended to solve the proposed model because it avoids the explicit calculation of probability distributions. The space complexity of the algorithm is only linear in the number of RE states and is independent of the enormous (four-dimensional) state space of the Markov process. Its time complexity is a linear function of the product of the queue capacity and the number of RE states. Our model is validated versus simulation estimates. The obtained conditional performance measures can accurately capture the queue accumulation and dissipation and reveal the effect of randomly changing environments. Numerical experiments provide some interesting findings. (1) The proposed stationary model coincides with the transient M( <q:math xmlns:q="http://www.w3.org/1998/Math/MathML" id="M5"> <q:mi>t</q:mi> </q:math> )/G <s:math xmlns:s="http://www.w3.org/1998/Math/MathML" id="M6"> <s:mfenced open="(" close=")" separators="|"> <s:mrow> <s:mi>x</s:mi> </s:mrow> </s:mfenced> </s:math> / <x:math xmlns:x="http://www.w3.org/1998/Math/MathML" id="M7"> <x:mi>C</x:mi> </x:math> / <z:math xmlns:z="http://www.w3.org/1998/Math/MathML" id="M8"> <z:mi>C</z:mi> </z:math> fluid queuing model under special conditions. (2) Under high traffic intensities, increasing the randomness in the duration time of the RE state leads to an obvious growth in the conditional queue length. (3) An increase in the facility length leads to an increase or a decrease in the average output rate, depending on whether the congestion dissipates effectively in one cycle. (4) A larger width is required to obtain the maximum average output rate for traffic demand with a greater nonuniformity.

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