Short CommunicationAnalysis of an MMPP/G/1/K queue with queue length dependent arrival rates, and its application to preventive congestion control in telecommunication networks
Introduction
Various queueing models have been studied for overload control to prevent congestion in telecommunication networks, in particular, ATM networks (Jain, 1996). ATM networks support diverse traffics with different service characteristics such as voice, data and video. These traffics are statistically multiplexed and transmitted in superhigh speed. Thus, unpredictable statistical fluctuation of traffic streams may cause congestion. An appropriate traffic control is required to prevent congestion and to gain bandwidth efficiency in ATM networks. Overload control is representative control scheme to prevent congestion.
Overload can be solved by controlling arrival rate or service rate. One representative overload control by service rate is well shown in papers (Choi and Choi, 1996, Choi et al., 1999, Sriram and Lucantoni, 1989). On the contrary, there are only a few papers about overload control by arrival rate. The partial buffer sharing is representative (Yegani, 1992). However, this model has only one threshold on buffer and assumes the arrival to be two-state MMPP.
We analyze a single-server queue with finite buffer and queue length dependent arrival rates. Arrivals of customers occur according to the queue length of buffer. We put appropriate thresholds on buffer and modulate arrivals. The arrival rates are determined according to whether the queue length at service completion epochs exceeds the thresholds or not. Arrivals are assumed to follow a Markov-modulated Poisson process (MMPP) by considering burstiness of arrival streams (Fischer and Meier-Hellstern, 1993).
We extend the paper of Yegani (1992) to the queueing system with multiple thresholds and general MMPP arrival. Overload control in our model makes the arrivals into network to be reduced if the congestion occurs (that is, if the queue length exceeds any threshold value). Thus, the arrival rate control by multiple thresholds can resolve the congestion faster. Finally, the network resources can be used efficiently, and more traffics can be allowed to share the bandwidth.
Our choice of MMPP is to model traffic streams with bursty characteristic and timecorrelation between interarrival times (Heffes and Lucantoni, 1996). Traffics such as voice and video in ATM networks have these properties. By using the embedded Markov chain method, we derive the queue length distribution at departure epochs. We also obtain the queue length distribution at an arbitrary time by the supplementary variable method. Finally, this information presents performance measures such as the loss probability and the mean waiting time.
With the rapid development of wireless communication networks, it is expected that fourth-generation mobile systems will be launched within decades. 4G mobile systems focus on seamless integrating the existing wireless technologies including GSM, wireless LAN, and Bluetooth. Contrasted with 3G, which merely focuses on developing new standards and hardware, 4G systems support comprehensive and personalized services, providing stable system performance and quality service (Hui and Yeung, 2003). Thus the success of 4G systems depends on the efficient traffic engineering, and the overload control plays a key role to prevent the congestion in the integrated network.
A description of the MMPP and preliminary for analysis is given in Section 2. In Section 3, we examine the queue length distribution at departure epochs and arbitrary time, and finally obtain performance measures such as loss and delay. Some numerical examples also are given in Section 4 to show effect of system parameters on performance measures.
Section snippets
Preliminary for analysis
Arrivals of customers follow an MMPP. The MMPP is a doubly stochastic Poisson process, in which the arrivals occur in a Poisson process with a rate that varies according to an underlying Markov process J(t). Let the Markov process J(t) have the state space {1, 2, … , N} and the matrix Q be its infinitesimal generator matrix:Concretely, arrivals in MMPP follow a Poisson process with rate λi whenever J(t) is in state i (1 ⩽ i ⩽ N). Then, the MMPP is said to have the
Analysis
Let Mr(t) (r = 1, 2, ⋯ , T) be the number of arrived customers according to Λr during the interval (0, t]. Now we define the conditional probabilitiesBy the Chapman–Kolmogorov’s forward equation, we have the following set of the differential-difference equations for the N × N matrix Pr(n, t) ≜ (pr(n, t)i,j)1⩽i,j⩽N,where Pr(−1, t) is the matrix 0.
Then, it is easily shown that the matrix Pr(n, t) has the probability
Numerical examples
In this section, we present some numerical examples to show the effects of the queue length dependent arrival systems on the loss probability and the mean waiting time. We set threshold values of an input process for numerical examples as L1 = 4 and L2 = 7. We assume that the arrivals of customers follow an MMPP withIn all numerical examples, we assume the service times S to be deterministic with mean μ = 1. We also take the buffer size K = 10, the MMPP
Conclusion
In this paper we analyzed an MMPP/G/1/K queueing system with queue length dependent arrival rates, and compared the system with threshold values with the system without threshold values in the numerical examples. The results of the paper can be applied for preventive congestion control in telecommunication networks such as ATM and 4G mobile system.
In this paper we showed that the performance of the system could be improved by adopting threshold values. It can be suggested for future research
References (8)
- et al.
The Markov-modulated Poisson process (MMPP) cookbook
Performance Evaluation
(1993) Congestion control and traffic management in ATM networks: Recent advances and a survey
Computer Networks and ISDN System
(1996)- et al.
The queueing system with queue length dependent service times and its application to cell discarding scheme in ATM networks
IEE Proceedings of Communications
(1996) - et al.
A queueing system with queue length dependent service times, with application to cell discarding in ATM networks
Journal of Applied Mathematics and Stochastic Analysis
(1999)