On the identification of Poisson arrivals in queues with coinciding time-stationary and customer-stationary state distributions

1983 ◽  
Vol 20 (4) ◽  
pp. 860-871 ◽  
Author(s):  
Dieter König ◽  
Masakiyo Miyazawa ◽  
Volker Schmidt
Keyword(s):  
Stationary State ◽  
Queueing Systems ◽  
Sufficient Conditions ◽  
Arrival Process ◽  
Loss Probabilities ◽  
Poisson Arrivals ◽  
Number Of Customers

For several queueing systems, sufficient conditions are given ensuring that from the coincidence of some time-stationary and customer-stationary characteristics of the number of customers in the system such as idle or loss probabilities it follows that the arrival process is Poisson.

On the identification of Poisson arrivals in queues with coinciding time-stationary and customer-stationary state distributions

1983 ◽  
Vol 20 (04) ◽  
pp. 860-871 ◽  
Author(s):  
Dieter König ◽  
Masakiyo Miyazawa ◽  
Volker Schmidt
Keyword(s):  
Stationary State ◽  
Queueing Systems ◽  
Sufficient Conditions ◽  
Arrival Process ◽  
Loss Probabilities ◽  
Poisson Arrivals ◽  
Number Of Customers

For several queueing systems, sufficient conditions are given ensuring that from the coincidence of some time-stationary and customer-stationary characteristics of the number of customers in the system such as idle or loss probabilities it follows that the arrival process is Poisson.

Queues with paired customers

1981 ◽  
Vol 18 (3) ◽  
pp. 684-696 ◽  
Author(s):  
Guy Latouche
Keyword(s):  
Queueing Systems ◽  
Arrival Process ◽  
Service Rate ◽  
Probability Vector ◽  
Simple Manner ◽  
Special Service ◽  
Poisson Arrivals ◽  
The Difference ◽  
Number Of Customers ◽  
Stationary Probability Vector

Queueing systems with a special service mechanism are considered. Arrivals consist of two types of customers, and services are performed for pairs of one customer from each type. The state of the queue is described by the number of pairs and the difference, called the excess, between the number of customers of each class. Under different assumptions for the arrival process, it is shown that the excess, considered at suitably defined epochs, forms a Markov chain which is either transient or null recurrent. A system with Poisson arrivals and exponential services is then considered, for which the arrival rates depend on the excess, in such a way that the excess is bounded. It is shown that the queue is stable whenever the service rate exceeds a critical value, which depends in a simple manner on the arrival rates. For stable queues, the stationary probability vector is of matrix-geometric form and is easily computable.

Queues with paired customers

1981 ◽  
Vol 18 (03) ◽  
pp. 684-696 ◽  
Author(s):  
Guy Latouche
Keyword(s):  
Queueing Systems ◽  
Arrival Process ◽  
Service Rate ◽  
Probability Vector ◽  
Simple Manner ◽  
Special Service ◽  
Poisson Arrivals ◽  
The Difference ◽  
Number Of Customers ◽  
Stationary Probability Vector

Queueing systems with a special service mechanism are considered. Arrivals consist of two types of customers, and services are performed for pairs of one customer from each type. The state of the queue is described by the number of pairs and the difference, called the excess, between the number of customers of each class. Under different assumptions for the arrival process, it is shown that the excess, considered at suitably defined epochs, forms a Markov chain which is either transient or null recurrent. A system with Poisson arrivals and exponential services is then considered, for which the arrival rates depend on the excess, in such a way that the excess is bounded. It is shown that the queue is stable whenever the service rate exceeds a critical value, which depends in a simple manner on the arrival rates. For stable queues, the stationary probability vector is of matrix-geometric form and is easily computable.

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.

Queueing output processes

1976 ◽  
Vol 8 (2) ◽  
pp. 395-415 ◽  
Author(s):  
D. J. Daley
Keyword(s):  
Queueing Systems ◽  
Second Order ◽  
Arrival Process ◽  
Single Server ◽  
Stationary Condition ◽  
Waiting Room ◽  
Stationary Point Process ◽  
Poisson Arrivals ◽  
Service Times ◽  
Markovian Systems

The paper reviews various aspects, mostly mathematical, concerning the output or departure process of a general queueing system G/G/s/N with general arrival process, mutually independent service times, s servers (1 ≦ s ≦ ∞), and waiting room of size N (0 ≦ N ≦ ∞), subject to the assumption of being in a stable stationary condition. Known explicit results for the distribution of the stationary inter-departure intervals {Dn} for both infinite and finite-server systems are given, with some discussion on the use of reversibility in Markovian systems. Some detailed results for certain modified single-server M/G/1 systems are also available. Most of the known second-order properties of {Dn} depend on knowing that the system has either Poisson arrivals or exponential service times. The related stationary point process for which {Dn} is the stationary sequence of the corresponding Palm–Khinchin distribution is introduced and some of its second-order properties described. The final topic discussed concerns identifiability, and questions of characterizations of queueing systems in terms of the output process being a renewal process, or uncorrelated, or infinitely divisible.

Some Pitfalls of Black Box Queue Inference: The Case of State-Dependent Server Queues

1993 ◽  
Vol 7 (2) ◽  
pp. 149-157 ◽  
Author(s):  
Sheldon M. Ross ◽  
J. George Shanthikumar ◽  
Xiang Zhang
Keyword(s):  
Queueing Systems ◽  
Work Load ◽  
Black Box ◽  
Arrival Process ◽  
Service Rate ◽  
Actual System ◽  
Sample Distribution ◽  
State Dependent ◽  
Service Times ◽  
Number Of Customers

In several queueing systems the service rate of a server is affected by the work load present in the system. For example, a teller at a bank or a checker at a check-out counter in a supermarket may change the service rate depending on the number of customers present in the system. But the service rate as a function of the number in the system can rarely be measured. Consequently, in a typical model of such a system it is assumed that the service rate is constant. Hence, such systems with a single stage are often modeled by GI/GI/c queueing systems with mutually independent arrival and service processes. Then the observed service times are used to find a sample distribution that will represent the distribution of the assumed i.i.d. service times. The purpose of this paper is to explore the effect of this black box queue inference (BBQI) in its ability to predict the performance of the actual system. In this regard, we have shown that when the arrival process is Poisson, if the servers react favorably [unfavorably] to higher work loads (i.e., if the server increases [decreases] the service rate as the number of customers in the system increases) then the BBQI predictions will be pessimistic [optimistic]. This result can be used to identify the server's attitude toward higher work load.

scholarly journals Performance Evaluation of the Priority Multi-Server System MMAP/PH/M/N Using Machine Learning Methods

Mathematics ◽  
2021 ◽  
Vol 9 (24) ◽  
pp. 3236
Author(s):  
Vladimir Vishnevsky ◽  
Valentina Klimenok ◽  
Alexander Sokolov ◽  
Andrey Larionov
Keyword(s):  
Machine Learning ◽  
High Efficiency ◽  
Performance Characteristics ◽  
New Method ◽  
Arrival Process ◽  
Priority Systems ◽  
Loss Probabilities ◽  
Number Of Customers ◽  
Multi Server ◽  
Stationary Probabilities

In this paper, we present the results of a study of a priority multi-server queuing system with heterogeneous customers arriving according to a marked Markovian arrival process (MMAP), phase-type service times (PH), and a queue with finite capacity. Priority traffic classes differ in PH distributions of the service time and the probability of joining the queue, which depends on the current length of the queue. If the queue is full, the customer does not enter the system. An analytical model has been developed and studied for a particular case of a queueing system with two priority classes. We present an algorithm for calculating stationary probabilities of the system state, loss probabilities, the average number of customers in the queue, and other performance characteristics for this particular case. For the general case with K priority classes, a new method for assessing the performance characteristics of complex priority systems has been developed, based on a combination of machine learning and simulation methods. We demonstrate the high efficiency of the new method by providing numerical examples.

Queueing output processes

1976 ◽  
Vol 8 (02) ◽  
pp. 395-415 ◽  
Author(s):  
D. J. Daley
Keyword(s):  
Queueing Systems ◽  
Second Order ◽  
Arrival Process ◽  
Single Server ◽  
Stationary Condition ◽  
Waiting Room ◽  
Stationary Point Process ◽  
Poisson Arrivals ◽  
Service Times ◽  
Markovian Systems

The paper reviews various aspects, mostly mathematical, concerning the output or departure process of a general queueing systemG/G/s/Nwith general arrival process, mutually independent service times,sservers (1 ≦s≦ ∞), and waiting room of sizeN(0 ≦N≦ ∞), subject to the assumption of being in a stable stationary condition. Known explicit results for the distribution of the stationary inter-departure intervals {Dn} for both infinite and finite-server systems are given, with some discussion on the use of reversibility in Markovian systems. Some detailed results for certain modified single-serverM/G/1 systems are also available. Most of the known second-order properties of {Dn} depend on knowing that the system has either Poisson arrivals or exponential service times. The related stationary point process for which {Dn} is the stationary sequence of the corresponding Palm–Khinchin distribution is introduced and some of its second-order properties described. The final topic discussed concerns identifiability, and questions of characterizations of queueing systems in terms of the output process being a renewal process, or uncorrelated, or infinitely divisible.

scholarly journals Retrial BMAP/PH/N Queueing System with a Threshold-Dependent Inter-Retrial Time Distribution

Mathematics ◽  
2022 ◽  
Vol 10 (2) ◽  
pp. 269
Author(s):  
Valentina I. Klimenok ◽  
Alexander N. Dudin ◽  
Vladimir M. Vishnevsky ◽  
Olga V. Semenova
Keyword(s):  
Exponential Distribution ◽  
Performance Indicators ◽  
Queueing System ◽  
Queueing Systems ◽  
Arrival Process ◽  
Ph Distribution ◽  
Retrial Queueing ◽  
Phase Type ◽  
Number Of Customers ◽  
Retrial Queueing Systems

In this paper, we study a multi-server queueing system with retrials and an infinite orbit. The arrival of primary customers is described by a batch Markovian arrival process (BMAP), and the service times have a phase-type (PH) distribution. Previously, in the literature, such a system was mainly considered under the strict assumption that the intervals between the repeated attempts from the orbit have an exponential distribution. Only a few publications dealt with retrial queueing systems with non-exponential inter-retrial times. These publications assumed either the rate of retrials is constant regardless of the number of customers in the orbit or this rate is constant when the number of orbital customers exceeds a certain threshold. Such assumptions essentially simplify the mathematical analysis of the system, but do not reflect the nature of the majority of real-life retrial processes. The main feature of the model under study is that we considered the classical retrial strategy under which the retrial rate is proportional to the number of orbital customers. However, in this case, the assumption of the non-exponential distribution of inter-retrial times leads to insurmountable computational difficulties. To overcome these difficulties, we supposed that inter-retrial times have a phase-type distribution if the number of customers in the orbit is less than or equal to some non-negative integer (threshold) and have an exponential distribution in the contrary case. By appropriately choosing the threshold, one can obtain a sufficiently accurate approximation of the system with a PH distribution of the inter-retrial times. Thus, the model under study takes into account the realistic nature of the retrial process and, at the same time, does not resort to restrictions such as a constant retrial rate or to rough truncation methods often applied to the analysis of retrial queueing systems with an infinite orbit. We describe the behavior of the system by a multi-dimensional Markov chain, derive the stability condition, and calculate the steady-state distribution and the main performance indicators of the system. We made sure numerically that there was a reasonable value of the threshold under which our model can be served as a good approximation of the BMAP/PH/N queueing system with the PH distribution of inter-retrial times. We also numerically compared the system under consideration with the corresponding queueing system having exponentially distributed inter-retrial times and saw that the latter is a poor approximation of the system with the PH distribution of inter-retrial times. We present a number of illustrative numerical examples to analyze the behavior of the system performance indicators depending on the system parameters, the variance of inter-retrial times, and the correlation in the input flow.

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