Tollbooth tandem queues with infinite homogeneous servers

2015 ◽  
Vol 52 (04) ◽  
pp. 941-961 ◽  
Author(s):  
Xiuli Chao ◽  
Qi-Ming He ◽  
Sheldon Ross
Keyword(s):  
Performance Measures ◽  
System Performance ◽  
Stochastic Ordering ◽  
Tandem Queues ◽  
Poisson Arrival ◽  
Arrival Processes ◽  
Poisson Arrivals ◽  
Number Of Customers ◽  
Steady State Solutions ◽  
Transient And Steady State

In this paper we analyze a tollbooth tandem queueing problem with an infinite number of servers. A customer starts service immediately upon arrival but cannot leave the system before all customers who arrived before him/her have left, i.e. customers depart the system in the same order as they arrive. Distributions of the total number of customers in the system, the number of departure-delayed customers in the system, and the number of customers in service at time t are obtained in closed form. Distributions of the sojourn times and departure delays of customers are also obtained explicitly. Both transient and steady state solutions are derived first for Poisson arrivals, and then extended to cases with batch Poisson and nonstationary Poisson arrival processes. Finally, we report several stochastic ordering results on how system performance measures are affected by arrival and service processes.

Tollbooth tandem queues with infinite homogeneous servers

2015 ◽  
Vol 52 (4) ◽  
pp. 941-961 ◽  
Author(s):  
Xiuli Chao ◽  
Qi-Ming He ◽  
Sheldon Ross
Keyword(s):  
Performance Measures ◽  
System Performance ◽  
Stochastic Ordering ◽  
Tandem Queues ◽  
Poisson Arrival ◽  
Arrival Processes ◽  
Poisson Arrivals ◽  
Number Of Customers ◽  
Steady State Solutions ◽  
Transient And Steady State

In this paper we analyze a tollbooth tandem queueing problem with an infinite number of servers. A customer starts service immediately upon arrival but cannot leave the system before all customers who arrived before him/her have left, i.e. customers depart the system in the same order as they arrive. Distributions of the total number of customers in the system, the number of departure-delayed customers in the system, and the number of customers in service at time t are obtained in closed form. Distributions of the sojourn times and departure delays of customers are also obtained explicitly. Both transient and steady state solutions are derived first for Poisson arrivals, and then extended to cases with batch Poisson and nonstationary Poisson arrival processes. Finally, we report several stochastic ordering results on how system performance measures are affected by arrival and service processes.

Monotonicity results for queues with doubly stochastic Poisson arrivals: Ross's conjecture

1991 ◽  
Vol 23 (1) ◽  
pp. 210-228 ◽  
Author(s):  
Cheng-Shang Chang ◽  
Xiu Li Chao ◽  
Michael Pinedo
Keyword(s):  
Queueing Systems ◽  
Loss Probability ◽  
Service Time Distribution ◽  
Doubly Stochastic ◽  
Virtual Waiting Time ◽  
Two Systems ◽  
Arrival Processes ◽  
Poisson Arrivals ◽  
Number Of Customers ◽  
Monotonicity Results

In this paper, we compare queueing systems that differ only in their arrival processes, which are special forms of doubly stochastic Poisson (DSP) processes. We define a special form of stochastic dominance for DSP processes which is based on the well-known variability or convex ordering for random variables. For two DSP processes that satisfy our comparability condition in such a way that the first process is more ‘regular' than the second process, we show the following three results: (i) If the two systems are DSP/GI/1 queues, then for all f increasing convex, with V(i), i = 1 and 2, representing the workload (virtual waiting time) in system. (ii) If the two systems are DSP/M(k)/1→ /M(k)/l ∞ ·· ·∞ /M(k)/1 tandem systems, with M(k) representing an exponential service time distribution with a rate that is increasing concave in the number of customers, k, present at the station, then for all f increasing convex, with Q(i), i = 1 and 2, being the total number of customers in the two systems. (iii) If the two systems are DSP/M(k)/1/N systems, with N being the size of the buffer, then where denotes the blocking (loss) probability of the two systems. A model considered before by Ross (1978) satisfies our comparability condition; a conjecture stated by him is shown to be true.

Monotonicity results for queues with doubly stochastic Poisson arrivals: Ross's conjecture

1991 ◽  
Vol 23 (01) ◽  
pp. 210-228 ◽  
Author(s):  
Cheng-Shang Chang ◽  
Xiu Li Chao ◽  
Michael Pinedo
Keyword(s):  
Queueing Systems ◽  
Loss Probability ◽  
Service Time Distribution ◽  
Doubly Stochastic ◽  
Virtual Waiting Time ◽  
Two Systems ◽  
Arrival Processes ◽  
Poisson Arrivals ◽  
Number Of Customers ◽  
Monotonicity Results

In this paper, we compare queueing systems that differ only in their arrival processes, which are special forms of doubly stochastic Poisson (DSP) processes. We define a special form of stochastic dominance for DSP processes which is based on the well-known variability or convex ordering for random variables. For two DSP processes that satisfy our comparability condition in such a way that the first process is more ‘regular' than the second process, we show the following three results: (i) If the two systems are DSP/GI/1 queues, then for all f increasing convex, with V (i), i = 1 and 2, representing the workload (virtual waiting time) in system. (ii) If the two systems are DSP/M(k)/1→ /M(k)/l ∞ ·· ·∞ /M(k)/1 tandem systems, with M(k) representing an exponential service time distribution with a rate that is increasing concave in the number of customers, k, present at the station, then for all f increasing convex, with Q (i), i = 1 and 2, being the total number of customers in the two systems. (iii) If the two systems are DSP/M(k)/1/N systems, with N being the size of the buffer, then where denotes the blocking (loss) probability of the two systems. A model considered before by Ross (1978) satisfies our comparability condition; a conjecture stated by him is shown to be true.

scholarly journals Research on Fuzzy Three Stage Tandem Queues

2019 ◽  
Vol 8 (6S4) ◽  
pp. 1478-1481
Keyword(s):  
Membership Function ◽  
Performance Measures ◽  
Service Time ◽  
System Performance ◽  
Arrival Time ◽  
General Procedure ◽  
Membership Functions ◽  
Tandem Queues ◽  
The Membership Function

This project intends a general procedure to obtain the membership function of the performance measures in three stage tandem queues when the inter-arrival time and service time are in fuzzy. The basic idea is to diminish the crisp queue into fuzzy by applying the α-cut approach technique. A pair of parametric program is formulated to depict that family of crisp queue in which the membership functions of the system performance are acquired.

STAFFING A SERVICE SYSTEM WITH NON-POISSON NON-STATIONARY ARRIVALS

2016 ◽  
Vol 30 (4) ◽  
pp. 593-621 ◽  
Author(s):  
Beixiang He ◽  
Yunan Liu ◽  
Ward Whitt
Keyword(s):  
Service System ◽  
Heavy Traffic ◽  
Arrival Rate ◽  
Service Level ◽  
Arrival Process ◽  
Performance Impact ◽  
Poisson Arrival ◽  
Arrival Processes ◽  
Poisson Arrivals ◽  
The Impact

Motivated by non-Poisson stochastic variability found in service system arrival data, we extend established service system staffing algorithms using the square-root staffing formula to allow for non-Poisson arrival processes. We develop a general model of the non-Poisson non-stationary arrival process that includes as a special case the non-stationary Cox process (a modification of a Poisson process in which the rate itself is a non-stationary stochastic process), which has been advocated in the literature. We characterize the impact of the non-Poisson stochastic variability upon the staffing through the heavy-traffic limit of the peakedness (ratio of the variance to the mean in an associated stationary infinite-server queueing model), which depends on the arrival process through its central limit theorem behavior. We provide simple formulas to quantify the performance impact of the non-Poisson arrivals upon the staffing decisions, in order to achieve the desired service level. We conduct simulation experiments with non-stationary Markov-modulated Poisson arrival processes with sinusoidal arrival rate functions to demonstrate that the staffing algorithm is effective in stabilizing the time-varying probability of delay at designated targets.

A single server Markovian queuing system with limited buffer and reverse balking

2021 ◽  
Vol 12 (7) ◽  
pp. 1774-1784
Author(s):  
Girin Saikia ◽  
Amit Choudhury
Keyword(s):  
Steady State ◽  
Performance Measures ◽  
Mutual Fund ◽  
System Size ◽  
Queuing System ◽  
Single Server ◽  
Insurance Company ◽  
Number Of Customers ◽  
Steady State Probabilities

The phenomena are balking can be said to have been observed when a customer who has arrived into queuing system decides not to join it. Reverse balking is a particular type of balking wherein the probability that a customer will balk goes down as the system size goes up and vice versa. Such behavior can be observed in investment firms (insurance company, Mutual Fund Company, banks etc.). As the number of customers in the firm goes up, it creates trust among potential investors. Fewer customers would like to balk as the number of customers goes up. In this paper, we develop an M/M/1/k queuing system with reverse balking. The steady-state probabilities of the model are obtained and closed forms of expression of a number of performance measures are derived.

Application of Statistical Quality Control in Testing Labs for Estimation of Machine Interference

2019 ◽  
pp. 77-98
Author(s):  
Nitin K. Mandavgade ◽  
Santosh B. Jaju ◽  
Ramesh R. Lakhe
Keyword(s):  
Performance Measures ◽  
System Performance ◽  
Testing Equipment ◽  
Statistical Quality Control ◽  
Test Results ◽  
Uncertainty Of Measurement ◽  
Testing Laboratories ◽  
Machine Interference ◽  
Analysis Of Results

The performance and maintenance of testing laboratories is a prime issue. The quality of coal test results not only depends on performance of individual results but it also depends on the performance of various tests in the same laboratory. Machine interference is a significant problem in many manufacturing system and testing equipment. The variation of results for testing equipment may be due to various factors which need to calculate the uncertainty of measurement to show the accuracy of the machine. In case of coal testing laboratory, the plant layout and surrounding environment affects the performance of the system. The machine interference comes under variable causes which may affect the result. This chapter proposes a methodology for constructing system performance measures, finding out the various factors responsible for variations in result. The chapter deals with estimation of machine interference existence using variable control chart approach for coal testing equipment. The analysis of results for such machine interference will be useful and significant for system designers and practitioners.

Performance Measures for Multimodal Transportation Systems

1996 ◽  
Vol 1518 (1) ◽  
pp. 85-93 ◽  
Author(s):  
Richard H. Pratt ◽  
Timothy J. Lomax
Keyword(s):  
Land Use ◽  
Performance Measurement ◽  
Travel Time ◽  
Performance Measures ◽  
System Performance ◽  
Transportation Systems ◽  
Regional Networks ◽  
Multimodal Transportation ◽  
Decision Context ◽  
Multimodal Systems

Transportation systems analyses have been evolving as the decision context for improvement projects and programs has changed. The increased emphasis on the movement of persons and goods, and a recognition of the importance of system performance measures that address the needs and interests of the audiences for mobility information, will result in a very different set of procedures for evaluating transportation and land use infrastructure and policies. Some of the key underlying concerns of performance measurement for multimodal systems are presented. Definitions are included for congestion, mobility, and accessibility that are used to guide the development of performance measures. Travel time–based measures are seen as the most readily understandable quantities, and examples are used to show how mobility can be measured for locations, corridors, transit analyses, and regional networks.

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